Wednesday
Wednesday, 8:30 – 9:00
WA01: Opening
Stream: Opening
Room: Fermat
Wednesday, 9:00 – 10:00
WB01: Plenary – Martine LABBE
Stream: Plenary
Room: Fermat
Chair(s):
Sonia Cafieri

Linear bilevel optimization: overview and recent results
Martine Labbé A bilevel optimization problem consists in an optimization problem in which some of the constraints specify that a subset of variables must be an optimal solution to another optimization problem. This paradigm is particularly appropriate to model competition between agents, a leader and a follower, acting sequentially. In this talk I will focus on the simplest bilevel problems, those that are linear. I will present the main characteristics, properties and algorithms for these problems. Then, I will discuss some recent results showing that these problems are already extremely challenging. This is a joint work with Thomas Kleinert, Martin Schmidt and Frank Plein.
Wednesday, 10:15 – 11:30
WC01: Analysis of Non Linear Algorithms I
Stream: Analysis of Non Linear algorithms
Room: Fermat
Chair(s):
Tobias Seidel

Duality and penalty function methods for a nonlinear programming problem
Andrei Dan Halanay, Miruna Beldiman We introduce a multiobjective nonlinear programming problem defined on a product of sigmaalgebras. First, using a partitioning scheme we propose a dual model for our problem and obtain duality results under different generalized univexity hypothesis. Then, considering a pseudometric corresponding to this context we study two resolution methods using an exponential penalty function and then a logarithmic one. We prove the convergence for these methods, closing with some examples. 
New Results on Superlinear Convergence of Classical QuasiNewton Methods
Anton Rodomanov, Yurii Nesterov We present a new theoretical analysis of local superlinear convergence of classical quasiNewton methods from the convex Broyden class. As a result, we obtain a significant improvement in the currently known estimates of the convergence rates for these methods. In particular, we show that the corresponding rate of the BroydenFletcherGoldfarbShanno (BFGS) method depends only on the product of the dimensionality of the problem and the logarithm of its condition number. 
Calculating local and global minima with quADAPT
Tobias Seidel, KarlHeinz Kuefer A classical solution approach to semiinfinite optimization is based on discretizing the semiinfinite index set. To reduce the number of discretization points, they can be chosen adaptively. Recently a new adaptive discretization method (quADAPT) has been introduced, which guarantees a quadratic convergence of the iterates under mild regularity assumptions. In this talk we establish conditions under which a limit point is a local minimum. We show that no such result is possible in the case of global minima. Finally, we introduce modifications to the algorithm that make a statement possible.
WC02: Advances in mathematical optimization for machine learning and data analysis – Part I
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Le Thi Khanh Hien

An adaptive subsampled Hessianfree optimization method for statistical learning.
Jeremy RIEUSSEC, Fabian Bastin, Jean LaprésChartrand, Loïc ShiGarrier We consider nonconvex statistical learning problems and propose a variable samplepath method, where the sample size is dynamically updated to ensure a decrease in the true objective function with high probability. We integrate this strategy in a subsampled Hessianfree trustregion method with truncated conjugate gradient, relying on outer product approximations. The approach is compared to various adaptive sample approximation algorithms and stochastic approximation methods popular in machine learning. The efficiency of the approach is illustrated on various large size datasets. 
An Inertial Newton Algorithm for Deep Learning
Camille Castera, Jerome Bolte, Cédric Févotte, Edouard Pauwels We introduce a new secondorder inertial optimization method for machine learning. It exploits the geometry of the loss function while only requiring stochastic approximations of the function values and the generalized gradients. The algorithm combines both gradientdescent and Newtonlike behaviors as well as inertia. We prove the convergence of the algorithm for most deep learning problems using tame optimization and dynamical systems theory. Additionally we discuss and address the existence of spurious stationary points when using minibatch methods for nonsmooth problems. 
An Inertial Block Majorization Minimization Framework for Nonsmooth Nonconvex Optimization
Le Thi Khanh Hien, Duy Nhat Phan, Nicolas Gillis We introduce TITAN, an inertial block majorization minimization framework for nonsmooth nonconvex optimizatioN problems. TITAN is a block coordinate method that embeds inertial force to each majorizationminimization step of the block updates. The inertial force is obtained via an extrapolation operator that subsumes heavyball and Nesterovtype accelerations for block proximal gradient methods as special cases. We study subsequential convergence as well as global convergence for the generated sequence of TITAN. We illustrate the effectiveness of TITAN on the matrix completion problem.
WC03: DerivativeFree Optimization I
Stream: DerivativeFree Optimization
Room: Nash
Chair(s):
Massimo Roma

Regret Bounds for NoiseFree Bayesian Optimization
Sattar Vakili Bayesian optimisation is a powerful method for nonconvex blackbox optimization in low data regimes. However, the question of establishing tight performance bounds for common algorithms in the noiseless setting remains a largely open question. We establish new and tightest bounds for two algorithms, namely GPUCB and Thompson sampling, under the assumption that the objective function is smooth in terms of having a bounded norm in a Matérn RKHS. Furthermore, we do not assume perfect knowledge of the kernel of the Gaussian process emulator used within the Bayesian Optimization loop. 
On some Bayesian optimization recipes applied to monomultifidelity constrained black box problems
Nathalie Bartoli, Thierry Lefebvre, Youssef Diouane, Sylvain Dubreuil, Joseph Morlier This work aims at developing new methodologies to optimise computational costly complex systems (e.g., aeronautical engineering systems). The proposed surrogatebased method (often called Bayesian Optimization) uses adaptive sampling to promote a tradeoff between exploration and exploitation. Our inhouse implementation, called SEGOMOE, handles a high number of design variables and nonlinearities by combining mixtures of experts (local surrogate models ) for the objective and/or the constraints. An extension to multifidelity is also included when a variety of information is available. 
On the use of DerivativeFree Optimization in design Discrete Event Simulation models for Emergency Departments
Massimo Roma, Alberto De Santis, Tommaso Giovannelli, Stefano Lucidi, Mauro Messedaglia The most widely used tool for studying patient flow through an Emergency Department (ED) is Discrete Event Simulation (DES). However, to achieve high reliability of such a DES model, an accurate calibration procedure is firstly required in order to determine a good estimate of the model input parameters. In this work we propose a SimulationOptimization approach integrating DES models with a DerivativeFree Optimization method in order to determine the best values of model input parameters. The approach we propose, has been widely experimented on a large ED in Rome.
WC04: Optimisation in Aerospace Engineering
Stream: Optimisation in Aerospace Engineering
Room: Lagrange
Chair(s):
Edmondo Minisci

Minimumfuel Lowthrust EarthMoon Transfers in the SunEarthMoon System
Richard EPENOY In this work, a minimumfuel optimal control problem will be stated subject to the dynamics of the SunEarthMoon Bicircular Restricted FourBody Problem. To overcome the huge sensitivity of the problem, a massive derivativefree exploration will be used to find the zeros of the associated shooting function. Different families of mediumduration transfers will be identified so as longduration trajectories exploiting the counterparts of invariant manifolds defined in both SunEarth and EarthMoon Circular Restricted ThreeBody Problems leading to socalled lowenergy transfers. 
Analysis of Nature Inspired Methods for Interplanetary Trajectory Problems
Edmondo Minisci, Gabriela Ochoa Many nature inspired optimisers have been used in the past decades to solve interplanetary trajectory problems. These problems are thought to have funnel or multi funnel objective function landscapes, but only few relevant works have tried to properly characterise the landscape and, more importantly, link the shape of the objective functions with the performance of the metaheuristics and heuristics. To contribute to this research area, a Local Optima Networks will be used to analyse, visualise and compare the behaviour of different types of solvers. 
Interoperating direct and indirect solvers in optimal control
Olivier Cots, JeanBaptiste Caillau, Pierre Martinon Methods for solving optimal control problems fall into two main categories. Indirect methods based on Pontrjagin Principle are fast and accurate but require more work, in terms of mathematical analysis and a priori knowledge of structure of the solution, than direct transcription approaches that offer a good tradeoff between robustness and accuracy. For challenging problems, one may start with a direct method to find a rough solution, then refine it through an indirect method. We discuss this combined approach on applications and the interfacing between two solvers: Bocop and HamPath.
WC05: Variational Methods in Vector and Set Optimization
Stream: Multiobjective Optimization
Room: Pontryagin
Chair(s):
César Gutiérrez, Gabriele Eichfelder

Stability of set optimization problems
Ruben Lopez, Elvira Hernández We study the stability of set optimization problems when the data (feasible sets and objective maps) admit variations. The data vary by means of a variational convergence notion for setvalued maps. 
A Vectorization Scheme for Nonconvex Set Optimization Problems
Stefan Rocktäschel, Gabriele Eichfelder, Ernest Quintana In this talk, we examine a solution approach for setvalued optimization problems with respect to the lower less relation. Thereby, we formulate a parametric vector optimization problem whose solution set approximates, in a specific sense, that of the setvalued optimization problem with arbitrary accuracy. We also investigate particular classes of setvalued mappings for which the corresponding set optimization problem is equivalent to the vector optimization problem. Surprisingly, setvalued mappings with a convex graph are one of the classes for which this equivalence holds. 
Ekeland variational principles in vector equilibrium problems
César Gutiérrez The talk addresses Ekeland variational principles for vector bifunctions defined on complete metric spaces. Several versions of that result are stated via a scalarization approach and new lower semicontinuity and lower boundedness assumptions. As a result, some recent Ekeland variational principles of the literature are improved since weaker hypotheses are assumed.
WC06: Applications of Optimization I
Stream: Applications of Optimization
Room: Moreau
Chair(s):
Michal Kocvara

Topological optimization of porous architectural materials using a multiphysics homogenization method
Godfred AGYEKUM OHENEBA, Laurent Cangemi, François JOUVE We will present a topological optimization method based on a homogenized formulation of poroelastic type, of a fluid mass transfer problem within a material with periodically controlled microstructure.The optimization consists in minimizing an energytype objective function, whose physical properties are correlated with a density of matter and with a microstructure at any point of the domain.Examples of optimized design results will be presented. 
A PrimalDual Augmented Lagrangian Algorithm for SDPs in truss topology design
Arefeh Kavand In this talk, we focus on the augmented Lagrangian framework based on the PENNON algorithm. We suggest a primaldual approach with a number of useful modifications, for solving large SDP problems with lowrank solutions. For this, we rephrase Newton systems in the PENNON algorithm by primaldual systems which are then solved approximately by the preconditioned conjugate gradient method using newly designed preconditioners. The efficiency of the algorithm is demonstrated by numerical experiments for truss topology optimization problems with growing dimension. 
An InteriorPoint Method for LowRank Semidefinite Programming with Application to Truss Topology Design
Soodeh Habibi, Michal Kocvara General algorithms and software for semidefinite optimization are unsuitable to solve very large dimensional problems. In this talk, we focus on SDP problems with lowrank solutions. Within a standard framework of an interiorpoint method, we solve the linear systems by a preconditioned conjugate gradient method. We present a new preconditioner tailored to the lowrank structure of the solution. We solve large to very largescale SDP problems from structural optimization with either rankone or approximate lowrank solutions. In both cases, our Matlab code outperforms available SDP software.
Wednesday, 11:45 – 13:00
WD01: Analysis of Non Linear Algorithms II
Stream: Analysis of Non Linear algorithms
Room: Fermat
Chair(s):
Vladimir Shikhman

An Interior PointProximal Method of Multipliers for Convex Programming
Spyridon Pougkakiotis, Jacek Gondzio In this talk, we present an infeasible Interior Point Method (IPM) combined with the Proximal Method of Multipliers (PMM). The resulting algorithm (IPPMM) is interpreted as a primaldual regularized IPM, suitable for solving convex programming problems. Given this framework, we prove polynomial complexity of the algorithm for a wide class of problems, under standard assumptions. We derive a robust tuning for the penalty parameters as well as an illposedness detection mechanism. The method is implemented and its reliability is demonstrated through extensive experimentation. 
Topological approach to mathematical programs with switching constraints
Vladimir Shikhman We study mathematical programs with switching constraints (MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the Wstationary point set, continuous deformation of lower level sets can be performed. However, when passing a Wstationary level, the topology of the lower level set changes via the attachment of a wdimensional cell. The dimension w depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of biactive switching constraints. 
Local convergence of tensor methods
Nikita Doikov, Yurii Nesterov In this talk, we study local convergence of highorder Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitzcontinuous highorder derivative. The convergence both in function value and in the norm of minimal subgradient is established. Also, we show how local convergence of the methods can be globalized using the inexact proximal iterations.
WD02: Advances in mathematical optimization for machine learning and data analysis – Part II
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Dimitri Papadimitriou

A Bregman Method for Structure Learning on Sparse Directed Acyclic Graphs
Manon Romain We develop a Bregman proximal gradient method for structure learning on linear structural causal models. While the problem is nonconvex, has high curvature and is NPhard, Bregman gradient methods allow us to neutralize at least part of the impact of curvature by measuring smoothness against a highly nonlinear kernel. This allows the method to make longer steps and significantly improves convergence. Each iteration requires solving a Bregman proximal step which is convex and efficiently solvable for our particular choice of kernel. We test our method on various synthetic and real data. 
An adaptive, accelerated algorithm for stochastic constrained convex optimization
Ali Kavis, Kfir Levy, Francis Bach, Volkan Cevher We propose a novel adaptive, accelerated algorithm for stochastic constrained convex optimization. Our method, inspired by the MirrorProx method, simultaneously achieves the optimal rates for smooth/nonsmooth problems with either deterministic/stochastic firstorder oracles. This is done without any prior knowledge of the smoothness nor the noise properties of the problem. To the best of our knowledge, this is the first adaptive, unified algorithm that achieves optimal rates in the constrained setting. Through extensive numerical experiments, we demonstrate the performance of our framework. 
Recent Advanced in the Theory of (Adaptive) Stochastic Gradient Methods
Vivak Patel Stochastic Gradient Descent (SGD) and its adaptive variants are foundational algorithms in data science, where it is essential to understand the behavior of these methods. In this talk, we establish several novel results. For standard SGD, we demonstrate L2 and strong convergence for a broad class of nonconvex functions, subsuming most recent results. Moreover, we provide a unified consistency and asymptotic normality analysis for a broad class of convex functions and adaptive methods, improving on many recent efforts.
WD03: Interfaces between Optimization and Game Theory
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s):
Laura Rosa Maria Scrimali

Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems
Veronica Piccialli, Giampaolo Liuzzi, Marco Locatelli, Stefan Rass In this work, we address a game theory problem arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies that minimize a linear payoff functional. In this paper an additional quadratic term is added to the minimization problem representing switching costs, i.e., the costs for the defender of switching from a given strategy to another one. The resulting problem is a nonconvex QP problem with linear constraints, that is NPhard. We propose an effective branch and bound for solving it. 
Decentralize and Randomize: Faster Algorithm for Wasserstein Barycenters
Pavel Dvurechensky We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primaldual stochastic gradient method 
Solving nonmonotone equilibrium problems via a DIRECTtype approach
Mauro Passacantando, Stefano Lucidi, Francesco Rinaldi A global optimization approach for solving nonmonotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed approach is a combination of an improved version of the DIRECT algorithm with local minimizations. Unlike most existing solution methods for EPs, no monotonicitytype condition is assumed in this paper. Preliminary numerical results on several classes of EPs show the effectiveness of the approach.
WD04: Applications of Optimization II
Stream: Applications of Optimization
Room: Lagrange
Chair(s):
Sebastien Bourguignon

Continuous optimization of Fourier sampling schemes for MRI
Alban Gossard, Frédéric de Gournay, Pierre Weiss Fourier sampling is a critical issue in image processing. While Shannon and compressed sensing theories dominated the field for decades, a recent trend is to optimize the sampling scheme for a specific dataset. In this work, we’ll focus on methods that continuously optimize the sampling points with respect to a reconstruction algorithm and to a database. Most attempts in this direction report optimization issues. We show that this phenomenon cannot be avoided since the problem is intrinsically combinatorial and has a huge number of spurious minimizers. We propose ideas to leverage this issue. 
GPUAccelerated Plan Optimization for IntensityModulated Radiotherapy
Juan José Moreno Riado, Janusz Miroforidis, Dmitry Podkopaev, Ernestas Filatovas, Ignacy Kaliszewski, Ester M Garzon IntensityModulated Radiotherapy (IMRT) is a technique for cancer treatment that allows precise control over the geometry and intensity profile of radiation beams. Since the dose deposited in the patient’s body by a given IMRT plan is modeled by a sparse matrix, most of the computation time during the plan optimization stage is spent in sparse matrix multiplications. Therefore, an adequate use of HPC techniques can drastically reduce planning time. In this work, we describe a GPUaccelerated gradientbased optimization technique capable of generating adequate plans in a limited time span. 
Optimization of molecular descriptors using memetic algorithms
Savíns Puertas Martín, Juana Lopez Redondo, Horacio PérezSánchez, Pilar M. Ortigosa One of the aims of Drug Discovery is to find compounds similar to a reference. For this purpose, different methods are appearing, among which we highlight Virtual Screening (VS). Usually, the VS techniques are adhoc methods that optimize a particular descriptor used as a scoring function. In this work, we propose a generic memetic optimization algorithm able to deal with any descriptor. Results show that our proposal outperforms the solutions provided by the stateoftheart algorithms, previously proposed in the literature.
WD05: Polynomial Optimization I
Stream: Polynomial Optimization
Room: Pontryagin
Chair(s):
Marek Tyburec

Sparse momentsumofsquares relaxations for nonlinear dynamical systems with guaranteed convergence
Corbinian Schlosser, Milan Korda We present a sparse momentsumofsquares approach for sparse dynamical systems that can be applied to the problems: region of attraction, maximum positively invariant set and global attractor. We prove a decomposition of these sets provided that the vector field and constraint set posses certain structure. We combine these decompositions with existing methods that are based on infinitedimensional linear programming. In case of sparse polynomial dynamics, we show that these methods admit a sparse sumofsquares (SOS) approximation with guaranteed convergence. 
Polynomial optimization applied to set computation
Benoît Legat, Raphaël M. Jungers The search for a set satisfying given properties is commonly narrowed down to polytopes or ellipsoids. However, in high dimension, the size of the representation of feasible polytopes might not be practical and the restriction to ellipsoids might be too conservative. In this talk, we explore the computation of sets defined by polynomial functions. For convex sets, we discuss the choice to parametrize the search in terms of the gauge or support function of the set depending on the properties that the set should satisfy. 
Exploiting constant trace property in largescale polynomial optimization
Ngoc Hoang Anh Mai, Jean Bernard Lasserre, Victor Magron We prove that every semidefinite moment relaxation of a polynomial optimization problem (POP) with a ball constraint can be reformulated as a semidefinite program involving a matrix with constant trace property (CTP). As a result such moment relaxations can be solved efficiently by firstorder methods that exploit CTP, e.g., the conditional gradientbased augmented Lagrangian method. The efficiency and scalability of our framework are illustrated on secondorder moment relaxations for various randomly generated QCQPs. This is joint work with Lasserre, Magron and Wang.
WD06: Energetic and Environmental Applications of Optimization I
Stream: Energetic and Environmental Applications of Optimization
Room: Moreau
Chair(s):
Juana Lopez Redondo

Decomposition of longterm investment optimization models for largescale integration of wind power in Europe
Caroline Granfeldt To capture the variability in electricity generation from variable renewable energy sources, mathematical modeling of future electricity systems must include a fine discretization of time. However, this leads to hugescale optimization problems. We have developed a linear optimization model that maintains a high temporal resolution while capturing the most important aspects of the problem to minimize future costs of electricity production. I will discuss how a variable splittingbased Lagrangean relaxation and subgradient algorithm enables a parallel solution process for this model. 
ETS, Emissions and the EnergyMix Problem
Paolo Falbo We study the impact of ETS on emissions and energymix through a bilevel model. At the upper level, a policymaker maximizes a welfare function deciding the number of allowances to distribute to the electricity producers. At the lower level, two large producers decide the longterm capacity expansion among three technologies: renewables, coal and gas. The uncertainty is modelled through Markov chain bootstrapping scenarios, made of coal and gas prices and electricity demand. The problem is solved considering a large set of Pareto efficient solution between the two electricity producers. 
Optical characterization of heliostat facets through computational optimization
N.C. Cruz, R. Monterreal, Juana Lopez Redondo, J. FernándezReche, R.E. Orts, Pilar M. Ortigosa Solar power tower plants use mirrors, known as heliostats, to collect solar radiation. They consist of reflective panels, called facets, whose building specifications must be checked. The Direct Method studies the features of the facet’s surface itself. The Indirect Method uses special equipment, such as ultrasensitive cameras, to analyze the solar image reflected by the studied facet on the target. This work focuses on the latter approach and formalizes an optical characterization procedure through computational optimization to avoid the traditional trialanderror strategy.
Wednesday, 14:30 – 15:30
WE01: EUROPT Fellow 2020 Lecture
Stream: Plenary
Room: Fermat
Chair(s):
Laura Palagi, Giancarlo Bigi

Nonsmooth Optimization for Classification Problems in Machine Learning
Manlio Gaudioso Classification is a machine learning technique where the objective is to assign each sample of a given population to exactly one from among several possible classes. Samples are represented by numerical vectors in the space of a certain number of attributes (features) and a classifier is a mathematical tool that is designed starting from a dataset of samples with known class membership (the so called training set). Once the classifier has been built up, it is used to predict the class membership for newly incoming samples. Most of the times classification problems are binary, as the class considered are just two, and constructing the classifier requires solution of a numerical optimization problem to detect an “optimal” separation surface in the sample space. Effective classification methods (e.g. the classic Support Vector Machine one) are based on the use of separating hyperplanes, and the problem at hand is formulated as a (possibly large scale) quadratic program. Nonsmooth optimization comes into the play first when nonlinear separation surfaces are sought, thus we will survey polyhedral, spherical, ellipsoidal and conical separation models. In addition we will focus on some more complex classification problems, where nonsmooth objective functions are to be dealt with. We will consider, in particular: • The semisupervised setting, where class membership is assumed to be known just for a subset of the training set; • The multiple instance learning, where bags of samples instead of single ones are to be classified; • The feature selection process, where sparse classifiers are designed. In all the above cases the optimization problems to be tackled are nonconvex and nonsmooth. Very often, however, they can be put in the DC (Difference of Convex) form. We will also provide the results of some numerical experiments.
Wednesday, 15:45 – 17:30
WF01: ADMM, block variants and proximality
Stream: Alternating Direction Method of Multipliers and its Applications
Room: Fermat
Chair(s):
Jacek Gondzio

A block symmetric GaussSeidel decomposition theorem for convex quadratic programming and its applications in multiblock ADMM
KimChuan Toh For a multiblock convex composite quadratic programming (CCQP) with an additional nonsmooth term in the first block, we present a block symmetric GaussSeidel (sGS) decomposition theorem, which states that each cycle of the block sGS method is equivalent to solving the CCQP with an additional sGS proximal term constructed from the quadratic function. This theorem has played a key role in various recently developed inexact proximal ADMM for multiblock convex conic programming. We demonstrate how our sGSbased ADMM can be applied solve doubly nonnegative SDPs. 
A purely numerical linear algebra view on ADMM
Stefano Cipolla, Jacek Gondzio Because of its versatility and applicability in multiple fields, the nblock alternating direction method of multipliers (ADMM) for solving nonseparable convex minimization problems, has recently attracted the attention of many researchers. Despite the fact the connections between ADMM and GaussSeidel are well known, to the best of our knowledge, an analysis from the purely numerical linear algebra point of view is lacking in the literature. Aim of this talk is to present a series of very recent results obtained on this topic which shed further light on basic issues as convergence 
Primaldual proximal methods with Bregman distances
Xin Jiang, Lieven Vandenberghe We discuss Bregman distance extensions of the primaldual threeoperator (PD3O) and CondatVu proximal algorithms. When used with standard proximal operators these algorithms include several important methods, including ADMM, as special cases. Extensions to generalized Bregman distances are attractive if the complexity per iteration can be reduced by matching the Bregman distance to the structure in the problem. For practical implementation we introduce line search techniques in the proposed methods. The results will be illustrated with applications in semidefinite optimization. 
MultiBlock ADMM – Managing Randomness and Data Privacy in Optimization Algorithm Design
Yinyu Ye Randomization strategy has been widely used in designing optimization algorithms to achieve properties that deterministic algorithms cannot do, such as SGD, BCD, Reinforced Learning etc. In this talk, we show how randomized techniques help in greatly improving the efficiency of the multiblock alternating direction method with multipliers. On the other hand, we illustrate that too much randomness may be harmful to the algorithm convergence. Therefore, randomness needs to be carefully managed to take advantage of the strategy. In addition, we discuss how to implement ADMM while preserving data
WF02: Beyond Firstorder Optimization Methods for Machine Learning – Part I
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Fred Roosta, Albert Berahas

Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization
Albert Berahas, Frank E. Curtis Stochastic gradient and related methods for solving unconstrained stochastic optimization problems have been studied extensively in recent years. However, settings with general nonlinear constraints have received less attention, and many of the proposed methods resort to using penalty or Lagrangian methods, which are often not the most effective strategies. In this work, we propose and analyze stochastic optimization methods based on the sequential quadratic optimization methodology. We discuss advantages and disadvantages of our approaches. Collaborators: F. E. Curtis, D. Robinson & B. Zhou. 
Stochastic Polyak Stepsize for SGD: An Adaptive Learning Rate for Fast Convergence
Nicolas Loizou We propose a stochastic variant of the classical Polyak stepsize (Polyak, 1987) commonly used in the subgradient method. Although computing the Polyak stepsize requires knowledge of the optimal function values, this information is readily available for typical modern machine learning applications. Consequently, the proposed stochastic Polyak stepsize (SPS) is an attractive choice for setting the learning rate for stochastic gradient descent (SGD). We provide theoretical convergence guarantees for SGD with SPS in different settings, including strongly convex, convex and nonconvex functions. 
Systematic Secondorder Methods for Training, Designing, and Deploying Neural Networks
Amir Gholami Finding the right Neural Network model and training it for a new task requires considerable expertise and extensive computational resources. Moreover, the process often includes adhoc rules that do not generalize to different application domains. These issues have limited the applicability and usefulness of DNN models, especially for new learning tasks. This problem is becoming more acute, as datasets and models grow larger, which increases training time, making random/brute force search approaches quickly untenable. In large part, this situation is due to the firstorder stochastic gradient 
Distributed Learning of Deep Neural Networks using Independent Subnet Training
Anastasios Kyrillidis We propose a new approach to distributed neural network learning, called independent subnet training (IST). In IST, a neural network is decomposed into a set of subnetworks of the same depth as the original network, each of which is trained locally, before the various subnets are exchanged and the process is repeated. IST training has many advantages over standard data parallel approaches. We show experimentally that IST results in training time that are much lower than data parallel approaches to distributed learning.
WF03: Mixed Integer Non Linear Programming
Stream: Mixed Integer Non Linear Programming
Room: Nash
Chair(s):
Sourour Elloumi

Branchandbound algorithm applied to sparse optimization problems: a study of some exploration strategies.
Gwenaël Samain, Jordan Ninin, Sebastien Bourguignon Sparse optimization focuses on finding a solution to leastsquares problems with few nonzero components. Applications include inverse problems in signal processing, subset selection, or portfolio optimization. Optimization problems can be formulated as mixedinteger programs. A dedicated branchandbound algorithm is able to exploit the specific structure of such problems and finds the global minimum much faster than generic MIP solvers. We will present some results about the finetuning process of this branchandbound algorithm, focusing mainly on the exploration strategy. 
Continuous location in spaces with different lpnormed regions.
Moisés RodríguezMadrena, Martine Labbé, Justo Puerto In this work we address the singlefacility Weber location problem in a finite dimensional real space endowed with different lpnorms at different polyhedral regions. We propose an equivalent representation of the location problem as a mixedinteger second order cone mathematical programming formulation. Interior point algorithms embedded in a branchandbound search can be applied allowing to obtain approximate solutions with a high precision degree. Our approach is validated reporting some preliminary computational experiments. 
Largescale Global MixedInteger Nonlinear Problems Solver by a Bilevel GeneralizedCGRASP Metaheuristic Method
João Lauro Faco’, Ricardo Silva, Mauricio Resende Largescale MINLP problems where the numbers n of continuous bounded variables and m constraints are large, and d discrete variables, n+d>m can be solved in 2 levels. Repeat: (1) Optimization: a reduced problem in (nm) continuous variables and d discrete variables is solved by a GeneralizedCGRASP method where the random search and local improvement phases use a continuous and a discrete set. (2) Feasibility: test the other continuous variables feasibility solving a system of m NL equations by CGRASP, keeping fixed the former variables. If any variable violates a bound, do a changeofbasis. 
Tightest linear reformulations for polynomial optimization with binary variables
Mathieu Verchère, Sourour Elloumi Polynomial optimization problems with binary variables are known to be strongly NPhard. In this talk, we consider the reformulation of these problems into equivalent mixed integer linear problems. For a given polynomial with binary variables, many such linear reformulations may exist and we focus on the question of how to find a reformulation with a tight LP bound. For polynomials with degree up to four, we formulate this question as a mixed integer linear problem. We then provide insights on potential improvements, as well as a theoretical comparison with other existing methods.
WF04: Constrained optimization methods and solvers in Julia
Stream: Constrained optimization methods and solvers in Julia
Room: Lagrange
Chair(s):
Miguel F. Anjos

Krylov methods in interiorpoint algorithms: a computational survey
Mathieu Tanneau, Alexis Montoison Following growing interest in the community, we present a computational survey on the use of Krylov methods for solving the linear systems that arise in interiorpoint algorithms. First, we review the various linear systems that can be formulated from the socalled Newton system, and establish a nomenclature of which Krylov methods are suited to which linear system formulation. Then, we review a number of generic preconditioners, both from mathematical and practical perspectives. Finally, we compare the numerical behavior of each possible approach through extensive computational experiments. 
Hypatia.jl: A Generic Nonsymmetric Conic Optimization Solver in Julia
Chris Coey, Lea Kapelevich, Juan Pablo Vielma Hypatia is an opensource optimization solver in Julia and is accessible through a native interface and through JuMP/MathOptInterface. Hypatia makes it easy to model and solve primaldual conic problems involving general convex cones for which appropriate primal OR dual barrier functions are known. We introduce Hypatia’s interfaces and algorithms and demonstrate computational advantages of compact “natural” conic formulations over extended formulations that use only “classical” cones. We also describe some algorithmic advances that have helped to make Hypatia competitive. 
Conditional gradient methods for largescale constrained optimization
Mathieu Besançon, Sebastian Pokutta, Alejandro Carderera Conditional gradient algorithms allow the integration of convex constraints in a firstorder optimization method. We present a new Julia toolbox integrating several variants of the Conditional Gradient method and detail the design choices that enable largescale and atypical applications. In particular, we will present the linear minimization oracle interface making the library extensible and allowing users to leverage closedform solution of linear minimization subproblems when they are known. 
DiffOpt.jl differentiating your favorite optimization problems
Joaquim Dias Garcia, Mathieu Besançon, Benoît Legat, Akshay Sharma DiffOpt aims at differentiating optimization problems written in MathOptInterface (MOI). Because MOI is JuMP’s lower level interface with solvers, this will “just work” with JuMP problems. The current framework is based on existing techniques for differentiating the solution of optimization problems with respect to the input parameters. We will show the current state of the package that supports Quadratic Programs and Conic Programs. Moreover, we will highlight how other packages are used to keep the library generic and efficient.
WF05: Multiobjective Optimization: Uncertainty and Nonconvexity
Stream: Multiobjective Optimization
Room: Pontryagin
Chair(s):
Gabriele Eichfelder

A general branchandbound framework for global multiobjective optimization
Oliver Stein We develop a general framework for branchandbound methods in multiobjective optimization. Our focus is on natural generalizations of notions and techniques from the single objective case. After a review of the main ideas of singleobjective branchandbound we discuss the appropriate generalizations of upper and lower bounds, discarding tests, node selection and termination criteria to the multiobjective setting. As a novel tool for multiobjective branchandbound we introduce convergent enclosures for the set of nondominated points. Numerical results illustrate our approach. 
Extending the Robust (Relative) Regret Approach to a Multicriteria Setting
Patrick Groetzner, Ralf Werner Consider a multiobjective decision problem with uncertainty given as a set of scenarios. In the single critria case, a famous method is to compare the possible decisions under uncertainty against the optimal decision with the benefit of hindsight, i.e. to minimize the (possibly scaled) regret of not having chosen the optimal decision. In this talk, we extend the concept of regret to the multiobjective setting and introduce a proper definition of multivariate (relative) regret. In contrast to the existing approaches, we are not limited to finite uncertainty sets or interval uncertainty. 
Solving uncertain multiobjective optimization problems via epigraphical reformulations
Ernest Quintana, Gabriele Eichfelder In this talk, we consider a solution approach for setbased robust solutions to multiobjective optimization problems under uncertainty. Specifically, we derive a parametric family of (deterministic) semiinfinite multiobjective problems whose solution sets approximate, with desired accuracy, that of the original problem. The tractability of the semiinfinite constraints is also analyzed with tools of Fenchel duality. Our approach generalizes the standard epigraphical reformulation of robust scalar problems to the multiobjective setting. 
A New Directional Derivative for Set Optimization
Gabriele Eichfelder, Robert Baier, Tobias Gerlach Setoptimization has attracted increasing interest in the last years, as for instance uncertain multiobjective optimization problems lead to such problems with a setvalued objective function. Optimality conditions for these problems, for instance using directional derivatives, are still very limited. The key aspect for a useful directional derivative is the definition of the used set difference. We present a new set difference which avoids the use of a convex hull and which applies to arbitrary convex sets. It is based on the new concept of generalized Steiner sets.
WF06: Conic Optimization and related topics
Stream: Conic Optimization and related topics
Room: Moreau
Chair(s):
Paula Amaral

Minmax regret for Standard QPs and nonconvex fractional QPs
Immanuel Bomze, Paula Amaral, Patrick Groetzner Under finitely many uncertain scenarios, minmax regret for (possibly nonconvex) Standard QPs can be reduced to a minmax QCQP. We will follow this route and embed it into the more general framework for nonconvex fractional quadratic optimization with linear denominators, under linear (and/or also quadratic constraints) where copositivity bounds play an essential role to reduce the optimality gap in global optimization procedures. 
On Standard Quadratic Programs with Exact and Inexact Doubly Nonnegative Relaxations
E. Alper Yildirim, Yakup Görkem Gökmen We consider the doubly nonnegative (DN) relaxation of standard quadratic programs. We present a full algebraic characterisation of the set of instances of standard quadratic programs that admit an exact DN relaxation. This characterisation yields an algorithmic recipe for constructing such an instance. In addition, we explicitly identify three families of instances for which the DN relaxation is exact. We also provide an algebraic characterisation of the set of instances for which the DN relaxation has a positive gap and show how to construct such an instance using this characterisation. 
Copositive based bounds on the clique number
Paula Amaral, Immanuel Bomze For a given a graph G, the maximum clique problem consists in determining the size of the maximum complete subgraph, γ(G) in G. It is well known that determining γ(G) is an NPhard problem. This problem can be formulated as a copositive optimization problem. Copositivity detection is difficult; in particular, to decide if it is not copositive is NPcomplete. In this talk we present an algorithm that is based on the decomposition of a matrix in the doubly nonnegative cone. The algorithm gives upper and lower bounds on the clique number. We report computational results for a set of graphs. 
Optimal parameter tuning for pathfollowing methods in conic optimization
Anastasiia Ivanova Classical schemes for shortstep pathfollowing methods in conic optimization alternate a Newton step towards a target point on the central path and an increase of the parameter of the target point. The quantitative analysis of the performance of these methods relies on a compariso of the Newton decrement before and after the Newton step. We propose a novel scheme which replaces the Newton decrement with respect to a target point by an analog of the decrement with respect to the central path as a whole. Here the direction parallel to the central path is treated differently from the directions
Thursday
Thursday, 9:00 – 10:40
TA01: Nonsmoothness, inexactness and applications
Stream: Alternating Direction Method of Multipliers and its Applications
Room: Fermat
Chair(s):
Stefano Cipolla

Differentiating nonsmooth (minimisation or) saddle point problems
Antonin Chambolle In recent works with T.Pock, TU Graz, we adapted the “piggyback” method to compute the differential of a loss depending on the solution of a nonsmooth saddle point problem. The idea is to estimate the adjoint state by an iterative method parallel to the computation of the solution, rather than inverting of a system depending on the solution. One advantage is that it may be done also for degenerate problems where the inversion is impossible. While working in practice, the method is justified for smooth problems only. We will discuss attempts to generalize it to less smooth situations. 
Principled analysis of methods with inexact proximal computations
Mathieu BARRÉ, Adrien Taylor Proximal operations are among the most common primitives appearing in optimization. In particular, they are crucial tools in many standard splitting methods. We show that worstcase guarantees for algorithms relying on inexact proximal operations can be systematically obtained through a generic procedure based on semidefinite programming. This methodology is primarily based on the approach from (Drori & Teboulle, 2014) and on convex interpolation results, and allows producing nonimprovable worstcase analyzes. We illustrate it on various algorithms involving inexact proximal computations. 
A
Flexible Optimization Framework for Regularized Linearly Coupled
MatrixTensor Factorizations based on the Alternating Direction Method
of Multipliers
Jeremy Cohen, Carla Schenker, Evrim Acar Ataman Coupled matrix and tensor factorizations (CMTF) are frequently used to jointly analyze data from multiple sources, a task also called data fusion. We propose a flexible algorithmic framework for coupled matrix and tensor factorizations which utilizes Alternating Optimization (AO) and the Alternating Direction Method of Multipliers (ADMM). The framework facilitates the use of a variety of constraints, regularizations, loss functions and couplings with linear transformations in a seamless way. We demonstrate its performance on some simulated and real datasets. 
An
ADMMNewtonCNN Numerical Approach to a TV Model for Identifying
Discontinuous Diﬀusion Coeﬃcients in Elliptic Equations: Convex Case
with Gradient Observations
Xiaoming Yuan We consider a TV model for identifying the discontinuous diﬀusion coeﬃcient in an elliptic equation with observation data of the gradient of the solution, and show that the ADMM can be used effectively. We also show that one of the ADMM subproblems can be well solved by the activeset Newton method along with the Schur complement reduction method, and the other one can be eﬃciently solved by the deep convolutional neural network (CNN). The resulting ADMMNewtonCNN approach is demonstrated to be eﬃcient even for higherdimensional spaces with ﬁne mesh discretization.
TA02: Nonlinear Composite and Constrained Optimization – Part I
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Cong Bang Vu, Dimitri Papadimitriou

Proximal alternating minimization for solving discrete MumfordShah and AmborsioTortorelli models
Hoang Trieu Vy LE, Marion Foare, Nelly Pustelnik This work focuses on a class of imaging problems which can be reformulated as a nonlinear composite problem. We derive two proximal alternating minimization schemes with convergence guarantees to estimate critical points. In particular, we place our attention on the discrete MumfordShah and the AmborsioTortorelli models. We compare their behaviors when the discrete counterpart of the 1D Hausdorff measure is modeled either by an l1norm or AT penalization. The closedform expressions of the involved proximity operators are provided. 
Random extrapolation for primaldual coordinate descent
Ahmet Alacaoglu, Olivier Fercoq, Volkan Cevher We introduce a randomly extrapolated primaldual coordinate descent method that adapts to sparsity of the data matrix and the favorable structures of the objective function. Our method updates only a subset of primal and dual variables with sparse data, and it uses large step sizes with dense data. We prove linear convergence under metric subregularity for strongly convexstrongly concave problems, and almost sure convergence of the sequence and optimal sublinear convergence rates for the primaldual gap and objective values, in the general convexconcave case. 
PrimalDual Proximal Splitting Algorithms for LargeScale Convex Optimization
Laurent Condat, Adil Salim, Peter Richtarik We study algorithms to minimize a sum of convex functions, possibly composed with linear operators, using their individual gradient or proximity operator. Using two different constructions based on the splitting technique proposed by Davis and Yin in 2017, we recover existing algorithms and discover a new one, which we call PDDY. Different variants, distributed, decentralized, or with stochastic gradient, will be discussed. Moreover, we derive nonergodic sublinear or linear convergence rates, as well as new accelerated versions in presence of strong convexity, using varying stepsizes. 
On a PrimalDual Newton Proximal Method for Convex Quadratic Programs
Alberto De Marchi We introduce QPDO, a primaldual method for convex quadratic programs which builds upon the proximal point algorithm and a damped semismooth Newton’s method. The outer proximal regularization yields a numerically stable method, and we interpret the proximal operator as the unconstrained minimization of a suitable merit function. The inner minimization relies on linear systems that are always solvable and exact linesearch. We report on numerical results against stateoftheart solvers. QPDO proves to be a simple, robust, and efficient numerical method for convex quadratic programming.
TA03: Game theory, multilevel and dynamic optimization I
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s):
Mathias Staudigl

Consistent Multiscale Control of Stackelberg Games
Anna Thünen We present a Stackelberg game with a large number of followers and we also derive the mean field limit of infinitely many followers. The relation between optimization and meanfield limit is studied and conditions for consistency are established. Finally, we propose a numerical method based on the derived models and present numerical results. 
Approximation and exact penalization in hierarchical programming
Giancarlo Bigi, Lorenzo Lampariello, Simone Sagratella Hierarchical programs are optimization problems whose feasible set is the solution set of another problem. The talk focuses on lowerlevel problems that are nonparametric wrt upper level variables: minimization over the solution set of a variational inequality is considered, which is a peculiar semiinfinite program and encompasses simple bilevel and equilibrium selection problems. To tackle it, a suitable approximated version is introduced. This does not perturb the original program too much while allowing the exploitation of exact penalty approaches whose convergence properties are shown. 
Generalized Nash games with discontinuous cost functions: an new approach
Didier Aussel, David Salas, Kien Cao Van A generalized Nash equilibrium problem (GNEP) corresponds to a non cooperative interaction between a finite set of players in which the cost function and the feasible set of each player depend on the decisions of the other players. The classical existence result for generalized equilibria due to ArrowDebreu requires continuity of the cost functions. In this work, we provide an existence of solutions transferring this hypothesis to a “continuitylike” condition over the sublevel sets of the aforementioned functions. Comparison with Reny’s approach for discontinuous games is also considered. 
Qualitative stability for solution map of parameterized Nash equilibrium problem and application
Thanh Cong Lai Nguyen, Didier Aussel It is well known that the Nash Equilibrium Problem is a multiplayer model of noncooperative games. It is of interest to know whether the solution set of its parameterized form is stable or not. Here stability is understood in the sense of the semicontinuity of the solution setvalued map. We explore a number of different approaches based on either producttype assumptions or componentwise hypotheses. The qualitative stability is investigated under some properties such as semicontinuity, quasiconvexity of the cost functions of the players. Then application to singleleadermultifollower
TA04: Optimization under Uncertainty and Applications I
Stream: Optimization under Uncertainty and Applications
Room: Lagrange
Chair(s):
Eligius M.T. Hendrix

Kernel Distributionally Robust Optimization
JiaJie Zhu We propose kernel distributionally robust optimization (Kernel DRO) using insights from functional analysis. Our method uses reproducing kernel Hilbert spaces to construct ambiguity sets, which can be generalized to integral probability metrics and moment bounds. This perspective unifies existing robust and stochastic optimization methods. We prove a theorem that generalizes the classical duality in the mathematical problem of moments. Using universal RKHSs, our method applies to a broad class of loss functions, lifting common limitations such as the knowledge of the Lipschitz constant. 
Optimal residential Microgrid operation considering VehicletoHome (V2H) and VehicletoGrid (V2G) energy services
Energy Management System (EMS) plays a key role in operating Microgrids (MG). We consider a residential gridconnected MG equipped with PV system and PEV battery. Based on Stochastic Optimization and Machine Learning approaches, we investigate different scenarios of the MG optimization considering various uncertainties and constraints. The developed Home EMS (HEMS) optimizes the scheduling and the power dispatch of the MG considering V2G and V2H services and allows to identify household’s potential energy cost savings and the effectiveness of V2G and V2H strategies in operating the MG. 
Convex Optimization using Tunable FirstOrder Inexact Oracles
Guillaume Van Dessel, François Glineur Using blackbox firstorder methods with an approximate gradient, i.e. an inexact oracle, may significantly impact their convergence. In this work, we consider a tunable inexact oracle, which can provide firstorder information with increasing levels of exactness, albeit with an increasing computational cost, which happens e.g. where the objective function is the optimal value of an inner subproblem. Given a target final accuracy, we show how to select the inexactness level at each step in order to minimize total computational budget, and present several situations where this is beneficial.
Thursday, 11:00 – 12:40
TB01: Optimization for Air Transportation
Stream: Optimization for Air Transportation
Room: Fermat
Chair(s):
Andrija VIDOSAVLJEVIC

Optimization for safe urban air mobility
Mercedes Pelegrín, Claudia D’Ambrosio, Rémi Delmas, Youssef Hamadi Urban Air Mobility (UAM) will exploit the third dimension to smooth ground traffic in densely populated areas. Online automated air traffic management will be key to ensure safety and optimize airspace capacity. This work addresses the problem of Tactical Deconfliction in UAM, considering different sources of unexpected traffic disruptions. A mathematical programming model based on envisioned UAM corridors is proposed. Extensive computational experience will allow us to draw conclusions on the benefits and limitations of our approach. 
Trajectory optimization for the computation of safe emergency trajectories
Maëva ONGALEOBEYI, Damien Goubinat, Pierreloic Garoche, daniel delahaye Nowadays, during an unexpected event in flight, the crew has to compare all the alternatives and plan a trajectory, without any automatized decision support. This decision has to be executed quickly in a stressful environment. Thus, the creation of an emergency landing planner seems crucial. We want to create a planner capable of computing a trajectory that is : (a) safe, (b) flyable and (c) in accordance with the resources. For this purpose, we decided to combine three algorithms, providing each one of these features: a pathplanner, a path smoothing system and a continuous trajectory system. 
New formulation for aircraft conflict avoidance problem using Sequential convex MINLP technique
Renan Spencer Trindade, Claudia D’Ambrosio, Antonio Frangioni, Claudio Gentile Our work presents a new formulation for the aircraft conflict avoidance problem in twodimensional space, where only the change in flight direction angle is allowed. The goal is to find the minimal variation of the original aircraft route angle while respecting the aircraft’s minimum safety distance along the trip. We propose a formulation defined as the sum of nonconvex univariate functions of the original problem. Sequential Convex Mixed Integer NonLinear Programming is applied to the problem, defining a convex relaxation of the original problem. 
Twostage stochastic programming for the extended aircraft arrival management problem with multiple prescheduling points
Ahmed Khassiba, Fabian Bastin, Sonia Cafieri, Bernard Gendron, Marcel Mongeau We consider the twostage stochastic optimization problem where a set of aircraft, heading to a given airport, are to be prescheduled on a reference point in the neartoairport area in the first stage, and to be scheduled on the landing runway in the second stage. Actual arrival times on the prescheduling point are assumed to deviate randomly from target arrival times. In this work, we extend a previously proposed model to the case of multiple prescheduling points. Preliminary results are obtained on realistic instances from ParisCharlesdeGaulle airport.
TB02: Nonlinear Composite and Constrained Optimization – Part II
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Cong Bang Vu, Dimitri Papadimitriou

A primal dual method for optimization with nonlinear constraints
Mehmet Fatih Sahin We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constraints. We characterize the total computational complexity of our method subject to a verifiable geometric condition, which is closely related to the PolyakLojasiewicz and MangasarianFromowitz conditions. We also provide numerical evidence on machine learning problems, including the BurerMonteiro factorization of semidefinite programs. 
Safe Screening for the Generalized Conditional Gradient Method
Yifan Sun We explore the sparsity acquiring properties of a generalized conditional gradient method, where the constraint is replaced by a gauge penalty function. Without assuming bounded iterates, we show O(1/t) convergence of the function values and duality gap. We couple this with a safe screening rule, and show that at a rate O(1/(td^2)), the screened support matches the support at the solution, where d > 0 measures problem degeneracy. Numerical experiments support our theoretical findings. 
An Optimal Algorithm for Strongly Convex Minimization under Affine Constraints
Adil Salim, Laurent Condat, Dmitry Kovalev, Peter Richtarik Optimization problems under affine constraints appear in various areas of machine learning. We consider the task of minimizing a smooth strongly convex function F under the affine constraint L x = b, with an oracle providing evaluations of the gradient of F and multiplications by L and its transpose. We provide lower bounds on the number of gradient computations and matrix multiplications to achieve a given accuracy. Then we propose an accelerated primaldual algorithm achieving these lower bounds. Our algorithm is the first optimal algorithm for this class of problems. 
Larger stepsizes for some primaldual algorithms
Ming Yan, Zhi Li Many primaldual algorithms are developed to solve the optimization problem with a linear composition, and there was an common upper bound for the product of the primal and dual stepsizes. In this talk, I will present a large upper bound of the product for some primaldual algorithms. In addition, we provide examples to show that the upper bound is tight. Then we apply this upper bound to show the convergence of several decentralized algorithms under weaker conditions.
TB03: Solution Techniques for Variational Inequalities
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s):
Patrizia Daniele

Pseudomonotone variational inequalities: Dynamics and numerical schemes of Tseng type
Robert Csetnek We associate to a pseudomonotone variational inequality a Tsengtype dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into Tseng’s forwardbackwardforward algorithm with relaxation parameters, which we prove to convergence also when it is applied to the solving of pseudomonotone variational inequalities. In addition, we show that linear convergence is guaranteed under strong pseudomonotonicity. We close with numerical experiments which justify the theory presented. 
Variance Reduction schemes for stochastic Variational inequalities
Mathias Staudigl We develop a new stochastic algorithm with variance reduction for solving pseudomonotone stochastic variational inequalities. Our method builds on Tseng’s forwardbackwardforward algorithm, which is known in the deterministic literature to be a valuable alternative to Korpelevich’s extragradient method when solving variational inequalities over a convex and closed set governed by pseudomonotone, Lipschitz continuous operators. 
A second order dynamical system and its discretization for strongly pseudomonotone variational inequalities
Vuong Phan We consider a second order dynamical system for solving variational inequalities in Hilbert spaces. Under standard conditions, we prove the existence and uniqueness of strong global solution of the proposed dynamical system. The exponential convergence of trajectories is established under strong pseudomonotonicity and Lipschitz continuity assumptions. A discrete version of the proposed dynamical system leads to a relaxed inertial projection algorithm whose the linear convergence is proved. We discuss the possibility of extension to general monotone inclusions. 
Stochastic generalized Nash equilibrium seeking and Variational Inequalities
Barbara Franci, Sergio Grammatico We consider the stochastic generalized Nash equilibrium problem (SGNEP) in merely monotone games with expectedvalue cost functions and shared constraints. Specifically, we present a distributed SGNE seeking algorithm with a single proximal computation (e.g. projection) and one single evaluation of the pseudogradient mapping. Our scheme is inspired by the relaxed forwardbackward algorithm for variational inequalities by Malitsky (Mathematical programming, 2019) and convergence is proven under monotonicity of the pseudogradient, approximated with the average over a number of random samples.
TB04: Energetic and Environmental Applications of Optimization II
Stream: Energetic and Environmental Applications of Optimization
Room: Lagrange
Chair(s):
Catherine Choquet

MixedInteger Nonlinear Optimization for District Heating Network Expansion
Marius Roland, Martin Schmidt We present a MINLP for the expansion of district heating networks given a number of candidate consumers. Expansion decisions are made using the stationary state solution for the district heating system taking into account the pressure and thermal losses of the water pipes. We propose an approximation of the thermal energy equation and additional valid inequalities are derived to help global solvers solve the problem. Our numerical results show a tradeoff between connecting distant candidate consumers, thus increasing thermal and pressure losses, and the estimated average consumer payment. 
Optimal positioning of groundwater heat pumps using the mesh adaptive direct search algorithm
Smajil Halilovic, Fabian Böttcher, Thomas Hamacher Groundwater heat pumps (GWHP) are one of the main technologies for heating and cooling. Until today groundwater is not utilized systematically. A way to maximize the potential of the resource is to position GWHPs optimally. This problem is a PDE constrained optimization problem. Due to expensive gradient estimations, we take a derivativefree optimization approach using the mesh adaptive direct search algorithm (MADS). To reduce the computational load, several dynamic and static surrogates are used in numerical benchmarks with various numbers of heat pumps and applied to real case scenarios. 
Response Surface Optimization of Continuous Packed Bed Adsorption Processes for Textile Wastewater Treatment
Younes Abrouki, Abdelkader Anouzla, Hayat Loukili In this study, we investigated the response surface optimization of continuous packed bed adsorption processes for textile wastewater treatment using lowcost adsorbent developed from biowaste. Under the optimal conditions exposed by central composite design, it is possible to eliminate almost the majority of the pollutants present in textile wastewater. It was found that this process using continuous fixedbed adsorption column filled with biowastederived adsorbent was efficient at removing of Pollutants from textile Wastewater. 
Numerical validation of a game theory approach for the optimal control problem of groundwater pollution
Moussa Mory DIEDHIOU, Catherine Choquet We consider a spatial differential game modeling the noncooperation of two polluters for the optimal control of groundwater quality while maximizing the profit of their polluting activity. Spatiotemporal objectives are constrained by a hydrogeological model for the spread of the pollution in the aquifer which consists in a system of a nonlinear parabolic partial differential equation for the pollutant concentration coupled with an elliptic equation for the velocity of the flow. The problem under consideration belongs to the class of infinite dimensional multiobjective control problems.
TB06: Global Optimization
Stream: Global Optimization
Room: Moreau
Chair(s):
Leocadio G. Casado

Exact and heuristic algorithms for solving a MINLP sigle facility location and design problem for firm expansion
A firm has a given budget available to invest in a given geographical area where it already runs some other facilities. Competing facilities already exist in the market. In order to increase its profit, the firm can (1) locate a new facility, (2) change the quality of the existing facilities up or down or even close some of them (3) do both things. An MINLP formulation for this problem is introduced, and both an interval B&B algorithm and a heuristic procedure are proposed to cope with it. According to the results, small variations in the available budget may produce very different results. 
Nested branchandbound algorithm for minmax problem and constraints with quantifiers
Jordan Ninin, Dominique Monnet, Benoit Clement We consider global optimization problems of a minmax function subject to constraints with quantifiers. The goal is to minimize over x the maximum over y of f(x,y), and constraints with quantifiers must be satisfied for a set of value (for all y in S). We present an algorithm to address this nonsmooth objective function and the constraints with quantifiers in the same way. Our approach is a setvalued method based on two nested branchandbound algorithm, using interval analysis. This problem arises in structured robust control, semiinfinite optimization or minmax optimization. 
Towards accelerating global parameter estimation with reduced data sets
Susanne Sass, Angelos Tsoukalas, Ian H. Bell, Dominik Bongartz, Jaromil Najman, Alexander Mitsos Many large parameter estimation problems are intractable for deterministic global optimization solvers. With a Branch and Bound algorithm in mind, we therefore investigate whether valid lower bounds can be constructed based on reduced data sets. To that end, we take the equation of state for propane as a case study. Our results indicate that reduced models can successfully identify regions where the model based on the full data set yields high and low objective values. However, whether and to what extent there is a time gain due to data reduction depends on the actual data sample chosen. 
On simplex minds and monotonicity
Eligius M.T. Hendrix, Leocadio G. Casado, Frederic Messine, Boglárka G.Tóth Simplicial branch and bound type of methods could make use of gradient bounds in order to create novel bounds over a simplicial partition set. Our question is how to use ideas of Interval Arithmetic to generate novel bounds. Our second question is dealing with the concept of monotonicity and exploitation in a simplicial branch and bound framework. We will report on our findings thus far. This investigation has been supported by The Spanish Ministry (RTI2018437 095993B100), in part financed by the European Regional Development Fund (ERDF).
Thursday, 14:00 – 15:00
TC01: EUROPT Fellow 2021 Lecture
Stream: Plenary
Room: Fermat
Chair(s):
Miguel F. Anjos, Giancarlo Bigi

Sums of squares of polynomials and graphs
Monique Laurent We investigate a hierarchy of semidefinite bounds for the stability number alpha(G) of a graph G, that are based on its continuous copositive programming formulation: alpha(G) = min {t: t(I+A_G)J \in COP(n)}. Here, n is the number of vertices, A_G is the adjacency matrix, J is the allones matrix, and COP(n) is the copositive cone, which consists of the symmetric matrices M for which the polynomial p_M = sum_{i,j=1}^n M_{ij}x_i^2 x_j^2 is nonnegative over R^n. By replacing the copositive cone COP(n) by its inner conic approximations K^r, consisting of the matrices M for which the polynomial x^{2r}p_M is a sum of squares, we obtain the bounds theta^{r}(G), known to converge asymptotically to alpha(G). De Klerk and Pasechnik (2002) conjectured that finite convergence takes place at order r=alpha(G)1. However, even the weaker conjecture asking whether equality holds for some r is still open. We discuss old and new results around these questions and the cones K^r, which display a nice interplay between graph structure, polynomial optimization and real algebraic geometry. Based on joint work with Luis Felipe Vargas (CWI, Amsterdam).
Thursday, 15:15 – 16:30
TD01: Sparse and LargeScale Optimization
Stream: Sparse and LargeScale Optimization
Room: Fermat
Chair(s):
Emmanuel Soubies

On nondegenerate Mstationary points for mathematical programs with sparsity constraint
Sebastian Lämmel We study mathematical programs with sparsity constraint (MPSC) from a topological point of view. Special focus will be on Mstationary points from Burdakov et al. (2016). We introduce nondegenerate Mstationary points, define their Mindex, and show that all Mstationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic MPSC. Moreover, we discuss the issues of instability and degeneracy of points due to different stationarity concepts. 
AARBased Decomposition Method For Limit Analysis
Nima Rabiei, Ali Almisreb A method is suggested for decomposing a class of nonlinear convex programmes which are encountered in limit analysis. These problems have secondorder conic constraints and a single complicating variable in the objective function. The method is based on finding the distance between the feasible sets of the decomposed problems, and updating the global optimal value according to the value of this distance. The latter is found by exploiting the method of Averaged Alternating Reflections which is here adapted to the optimization problem at hand. 
Expanding Boundaries of Gap Safe Screening
Cassio Dantas, Emmanuel Soubies, Cédric Févotte Safe screening techniques allow the early elimination of zero coordinates in the solution of sparse optimization problems. In this work, we extend the existing Gap Safe screening framework by relaxing the global strongconcavity assumption on the dual cost function. Local regularity properties are considered instead. Besides extending safe screening to a broader class of functions that includes betadivergences (e.g., the KullbackLeibler divergence), the proposed approach also improves upon the existing Gap Safe screening rules on previously applicable cases (e.g., logistic regression).
TD02: Advances in DouglasRachford method – Part I
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Cong Bang Vu, Dimitri Papadimitriou

SplitDouglasRachford for composite monotone inclusions and SplitADMM
Luis BriceñoArias, Fernando Roldán We provide a generalization of the DouglasRachford splitting (DRS) for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method activates the linear operator separately from the monotone operators appearing in the inclusion and, in the simplest case when the linear operator is the identity, it reduces to classical DRS. Moreover, the weak convergence of primaldual sequences to a point in the extended solution set is guaranteed, generalizing Svaiter (2011). As in Gabay (1983), we derive a new SplitADMM in the convex optimization setting. 
Anderson Accelerated DouglasRachford Splitting
Anqi Fu, Junzi Zhang, Stephen Boyd We consider the problem of nonsmooth convex optimization with linear equality constraints, where the objective function is only accessible through its proximal operator. To solve it, we propose an Anderson accelerated DouglasRachford splitting (A2DR) algorithm, which we show either globally converges or provides a certificate of infeasibility/unboundedness. Applied to a block separable objective, A2DR partially decouples so its steps may be carried out in parallel, yielding an algorithm that is highly scalable. We describe an opensource implementation along with a variety of applications. 
DouglasRachford splitting and ADMM for nonconvex optimization: Accelerated and Newtontype algorithms
Lorenzo Stella, Andreas Themelis, Panagiotis Patrinos We propose a new linesearch extension to DouglasRachford splitting (DRS) and ADMM, that accelerates them with quasiNewton directions. The algorithms are suited for nonconvex problems, require the same blackbox oracle as the original methods, and maintain their (subsequential) convergence. Experiments show that using LBFGS and Anderson acceleration directions greatly improves convergence over vanilla DRS and ADMM, making them more robust to illconditioning. Under regularity and nondegeneracy assumptions at the limit point, superlinear convergence is shown when using Broyden directions.
TD03: Advances in Variational Inequalities and Equilibrium Problems
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s):
Robert Csetnek

Optimal emergency evacuation with uncertainties
Laura Rosa Maria Scrimali, Georgia Fargetta Emergency management after crises or natural disasters is a very important issue shared by many countries. In particular, evacuation planning is a complex and challenging process that takes into account several elements. For this reason, optimization tools are fundamental to make policy makers predict or evaluate different disaster scenarios. In this talk, we present an emergency evacuation model where a population has to be evacuated from crisis areas to shelters and propose an optimization formulation for minimizing a combination of the transportation cost and the transportation time. In add 
A stochastic variational approach to study economic equilibrium problem under uncertainty
Domenico Scopelliti, Monica Milasi This talk deals with an economic equilibrium problem under time and uncertainty. We suppose that the market evolves in a finite sequence of time periods; at final date different states of the world can occur. We first reformulate the equilibrium by means of a stochastic quasivariational inequality. In view of this characterization we can give some qualitative and quantitative properties of equilibrium solutions. Our approach can fit in several decisionmaking problems where the characteristcs elements are affected by uncertainty. 
A variational approach for international human migration networks with and without regulations
Patrizia Daniele, Anna Nagurney International human migration has transformed economies and societies. The new millennium, with climate change and its impacts, and increasing conflicts and displacements, has experienced a great increase in international migrants, with associated challenges faced by governments. We propose the modeling, analysis, and solution of international human migration problems by developing a network model with and without regulations. The formalism uses the theory of variational inequalities, coupled with Lagrange analysis.
TD04: Polynomial Optimization II
Stream: Polynomial Optimization
Room: Lagrange
Chair(s):
Corbinian Schlosser

Topology optimization of discrete structures by polynomial optimization
Marek Tyburec, Jan Zeman, Martin Kružík, Didier Henrion Although finding continuous crosssections of minimumcompliance frame structures is a fundamental problem of structural design, only local optimization techniques are adopted. Here, we develop a semidefinite programming formulation with a nonconvex polynomial matrix inequality constraint and solve it by the momentsumofsquares hierarchy. Besides lower bounds from the hierarchy, we recover a feasible solution in each relaxation, building a sequence of upper bounds. The gap between the bounds quantifies the solution quality and establishes a simple sufficient condition for global optimality. 
The approximation of measures by positive moment sequences
Lorenzo Baldi, Bernard Mourrain To solve Polynomial Opt. problems Lasserre introduced the Moment Matrix hierarchy, approximating measures with a convex cone of positive linear functionals, and proved the convergence of optimal value to the minimum in the Archimedean case. We analyse the convex cone of truncated positive moment sequences and show that truncated measures with low rank moment matrix are extremal rays. We show the convergence in Hausdorff distance of a convex section of the outer approximation to probability measures, and describe the order of convergence using an effective version of Putinar Positivstellensatz. 
RAPOSa: a polynomial optimization solver
Brais González Rodríguez, Julio GonzálezDíaz, Ángel Manuel González Rueda, Beatriz PateiroLópez, Diego Rodriguez Martinez, David R Penas In this talk we will present RAPOSa, a new solver for polynomial optimization problems based on the reformulation linearization techniques (originally introduced by Sherali and Tuncbilek in 1991). The current implementation incorporates most of the features discussed in past literature and other additional enhancements. The present version of the algorithm has proven to be very efficient, and comparisons with other global optimization solvers will be presented.
TD05: Optimal Control and Optimization in Economics, Finance and Management I
Stream: Optimal Control and Optimization in Economics, Finance and Management
Room: Pontryagin
Chair(s):
Ioannis Baltas

Stochastic differential games for optimal investment problems in a Markov regimeswitching jumpdiffusion market
GerhardWilhelm Weber, Emel Savku We apply dynamic programming principle to discuss two optimal investment problems by using zerosum and nonzerosum stochastic game approaches in a continuoustime Markov regimeswitching environment. The first application is a zerosum game between an investor and the market, and the second one formulates a nonzerosum stochastic differential portfolio game as the sensitivity of two investors’ terminal gains. We derive regimeswitching HamiltonJacobiBellmanIsaacs equations and obtain explicit optimal portfolio strategies with FeynmanKac representations of value functions. 
Random oligopolistic market optimal equilibrium control problem
Annamaria Barbagallo The aim of the talk is to analyze the policymaker’s point of view of the random oligopolistic market equilibrium problem and present a stochastic optimal control equilibrium problem. More precisely, we study a model in which control policies may be imposed to regulate the amounts of exportation in random way. Control policies are implemented by imposing higher taxes or subsidies in order to restrict or encourage the exportation. We prove that the system that controls the commodity exportations in random way is expressed by a stochastic inverse variational inequality. 
Optimal
management of Defined Contribution pension funds during the
distribution phase under the effect of inflation, mortality and
uncertainty
Ioannis Baltas, Athanasios Yannacopoulos, GerhardWilhelm Weber We study the problem of optimal management of defined contribution (DC) pension funds, during the distribution phase, within a model uncertainty framework by taking into account the effect of (i) inflation and (ii) mortality. By employing robust control and dynamic programming techniques, we provide closed form solutions for the case of the exponential utility function. Moreover, we provide a detailed study of the limiting behavior of the associated stochastic differential game. Finally, we present a novel numerical approach that elucidates the effect of robustness and inflation.
TD06: Applications of Optimization III
Stream: Applications of Optimization
Room: Moreau
Chair(s):
Jordan Ninin

Network models for the human migration phenomenon
Giorgia Cappello Largescale migration flows are posing immense challenges for governments around the globe. In this work we introduce and analyze different multiclass human migration network models under both useroptimizing and systemoptimizing approaches, using tools of VI and Game theory. Four scenarios are proposed: in one model we introduce policy intervention, in the form of subsidies; in another model the locations associated with migration are subject to capacities; in another one we introduce regulations inspired by the Covid19 pandemic. Some numerical examples are provided. 
An approach on efficiency measure for a fully fuzzy DEA
Manuel AranaJiménez In this work, we deal with a model of Data Envelopment Analysis (DEA) poblem whose inputs and outputs data can be given by means of fuzzy numbers, what derives a fully fuzzy DEA. We propose a new radial, inputoriented approach, based on arithmetic operations with fuzzy numbers, as well as LUfuzzy partial order. We propose a method with two phases to assess the relative efficiency of each Decision Making Unit (DMU), and classify each DMU. In each phase a fully fuzzy linear programming is formulated, which is transformed into a related multiobjective optimization problema. 
Toolpath optimization for freeform surfaces machining
mahfoud herraz, Marcel Mongeau, Mohammed Sbihi A twostep optimization procedure, specifically dedicated to toolpath planning for milling of freeform surfaces is presented. A full optimization model is built relying on stateoftheart blackbox optimizations methods. Because of specific industrial needs, this procedure has to include a clustering phase and a milling simulation phase. Furthermore, given the context, calculation time is very important. Therefore, a surrogate model, based on Principal Component Analysis is defined. The approach is benchmarked with several test case surfaces, and results are presented and discussed.
Thursday, 16:50 – 18:30
TE01: Optimization and Artificial Intelligence I
Stream: Optimization and Artificial Intelligence
Room: Fermat
Chair(s):
Fabian Bastin

Exploiting Sparse Decision Trees To Learn Minimal Rules
Tommaso Aldinucci Decision tree algorithms have been one of the most famous and used algorithms in machine learning since the early 1980’s. Given their intrinsic explainability, nowadays they are widely used in contexts where transparency is desired. In this work we propose a new heuristic strategy to extract minimal rules from decision trees. At first the algorithm builds a sparse structure then uses a local optimizer to improve the quality of the tree. We finally show that the rule set extracted from the final model can be often simplified using elementary rules of boolean logic. 
Improved Forward Selection Algorithms for Sparse Principal Component Analysis
Mustafa Pinar Sparse PCA is a wellknown NPHard problem where the principal components obtained are desired to be sparse. This is useful in various studies due to its interpretability and robustness. In literature, heuristics or SDP relaxations are used to solve this problem. In this study, we propose multiple novel heuristics using eigenvalue approximation such as Gershgorin Circle’s to improve the performance of greedy forward selection methods. We have observed that our methods perform better in underdetermined and high dimensional settings compared to the existent methods in the literature. 
Kernel Regression with Hard Shape Constraints
PierreCyril Aubin, Zoltán Szabó Regression problems typically involve shape constraints due to qualitative priors or physical constraints, such as positivity or monotonicity. This writes as an infinite number of pointwise inequalities. Dealing with reproducing kernel Hilbert spaces, I describe how to solve a strengthened problem with a finite number of secondorder cone constraints with bound guarantees, and application to joint quantile regression, to Engel’s law and to analyze vehicle trajectories. This approach is extended to linearquadratic optimal control problems with state constraints. 
Leveraging deep reinforcement learning through the framework of operations research
Ysaël Desage, Fabian Bastin, François Bouffard While research in the field of artificial intelligence has been booming in recent year, especially in the field of machine learning, most applications are still in their infancy. In this perspective, we propose a hybrid innovative approach combining the performance and neologism of deep reinforcement learning with the reliable empirical and proven application framework of classical operations research. More concretely, we detail the proposed methodology before presenting the results on different problems and sub instances related to the power grid and the optimization of its actors.
TE02: Beyond Firstorder Optimization Methods for Machine Learning – Part II
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Fred Roosta, Albert Berahas

Fulllow evaluation type methods for derivativefree optimization
Luis Nunes Vicente, Albert Berahas, Oumaima Sohab We propose a new class of methods for DFO that considers two types of iterations. The first is expensive in fevals, but performs well in the smooth, nonnoisy case. We considered BFGS computed over gradients approximated by finite differences. The second is cheap in fevals, more appropriate under noise or nonsmoothness. We considered probabilistic direct search with 2 random directions. The resulting method is globally convergent in the nonsmooth case and yields the appropriate rates in the smooth case. Numerical results showed that it is efficient and robust for all types of problems. 
NewtonMR Algorithms With Complexity Guarantees for Nonconvex Optimization
Fred Roosta Classically, the conjugate gradient (CG) method has been the dominant solver in most inexact Newtontype methods for unconstrained optimization. In this talk, we consider replacing CG with the minimum residual method (MINRES), which is often used for symmetric but possibly indefinite linear systems. We show that MINRES has an inherent ability to detect negativecurvature directions. Equipped with this, we discuss linesearch and trustregion variants of NewtonMR, which can be used for optimization of general nonconvex objectives, and that come with favourable complexity guarantees. 
Inexact Restoration with Subsampled Trustregion methods for finitesum minimization
Stefania Bellavia, Natasa Krejic, Benedetta Morini In this talk we will focus on subsampling second order trustregion procedures combined with the Inexact Restoration approach for finitesum minimization. The sample size of function approximations is computed through a deterministic rule inspired by the inexact restoration method and the trustregion step is either accepted or rejected using a suitable merit function. We discuss the local and global properties for finding approximate first and secondorder optimal points and show results from the numerical experience. 
Retrospective Approximation for Stochastic Optimization
Raghu Bollapragada We present Retrospective Approximation as a universal sequential sampleaverage approximation paradigm where during each iteration, a samplepath approximation problem is implicitly generated using an adapted sample size, and solved to an adapted error tolerance, using a “deterministic method” such as the line search quasiNewton method. The principal advantage of RA is that decouples optimization from stochastic approximation, allowing the direct adoption of existing deterministic algorithms without modification, thus mitigating the need to redesign algorithms for the stochastic context. 
Global optimization using random embeddings
Estelle Massart, Coralia Cartis, Adilet Otemissov We present a general random subspace algorithmic framework for global optimization and analyse its convergence using tools from conic integral geometry and random matrix theory. We then particularise this framework and analysis for the class of functions with low effective dimension. We show that its convergence does not depend on the ambient dimension, and are able to estimate the effective dimension in the run of the algorithm. Encouraging numerical results are also presented that use local or global solvers in the subspace.
TE03: Optimal Control and Applications I
Stream: Optimal Control and Applications
Room: Nash
Chair(s):
Theodore Trafalis

Optimal control problems on stratified spaces
Othmane Jerhaoui We are interested in control problems on stratified spaces. In such problems, the state variable space is partitioned in different manifolds with boundary glued together along their boundaries. Each manifold is associated with a compact control set, a controlled dynamics and a cost function. We are interested in analyzing the set of trajectories, and in characterizing the value function. Moreover, we will discuss some numerical schemes for such control problems. Finally, we discuss some generalizations of this problem on some metric spaces resembling this setting called Aleksandrov spaces. 
ZeroSum Deterministic Differential Game in Infinite Horizon with Continuous and Impulse Controls
HAFID LALIOUI We consider a zerosum deterministic differential game with two players adopting both continuous and impulse controls in infinite time horizon. The form of impulses supposed to be of general term (depends on non linear functions) and the cost of impulses being arbitrary (depends on the state of the system). We use the dynamic programming principle (DPP) and the viscosity solution approach to prove that the value function turns out to be, under Isaacs condition, the unique viscosity solution to the corresponding HamiltonJacobiBellmanIsaacs (HJBI) variational inequalities (QVIs). 
Mixed ZeroSum Stochastic Differential Game and Doubly Reflected BSDEs with a Specific Generator.
Nacer OURKIYA This paper studies the mixed zerosum stochastic differential game problem. We allow the functionals and dynamics to be of polynomial growth. The problem is formulated as an extended doubly reflected BSDEs with a specific generator. We show the existence of solution for this doubly reflected BSDEs and we prove the existence of a saddlepoint of the game. Moreover, in the Markovian framework we prove that the value function is the unique viscosity solution of the associated HamiltonJacobiBellman equation. 
LearningBased Nonlinear Hinfinity Control via GameTheoretic Differential Dynamic Programming
Wei Sun, Theodore Trafalis We present a learningbased nonlinear Hinf control algorithm that guarantees system performance under learned dynamics and disturbance estimate. The Gaussian Process regression is utilized to update the dynamics of the system and provide disturbance estimate based on data gathered by the system. A softconstrained differential game associated with the disturbance attenuation problem in nonlinear Hinf control is then formulated to obtain the nonlinear Hinf controller. The differential game is solved through the GameTheoretic Differential Dynamic Programming algorithm in continuous time.
TE04: Advanced Optimization Methods I
Stream: Advanced Optimization Methods
Room: Lagrange
Chair(s):
Marat Mukhametzhanov

ReducedOrder Parameter Optimization by Sequential ALIENOR method
Daniele Peri Space Filling Curves (SFC) are a class of curves that completely covers a portion of a Ndimensional space asymptotically: an arbitrary position of the space is then associated with a single parameter (the curved abscissa along the SFC). This way, an N dimensional optimization problem can be converted in a 1D problem (ALIENOR method). In this paper a sequential approach using SFC is presented, managing hundreds of design variables: solution is obtained by iterativerly focusing on smaller and smaller portions of the feasible set. An industrial application is also presented. 
Univariate Lipschitz Global Optimization Methods for Finding the Shape Parameter Value in Radial Basis Functions
Marat Mukhametzhanov, Yaroslav Sergeyev Radial basis function (RBF) interpolation is considered in this talk. It is wellknown that the value of the shape parameter of the RBFs has a strong impact on both the accuracy and stability of the results. It has been proposed recently to use univariate Lipschitz global optimization methods for finding an optimal value of the parameter. Locallybiased versions of the wellknown “Dividethebest” algorithms are presented for this purpose. Numerical experiments on several randomized and reallife test problems show the advantages of proposed the techniques. 
On a new shape derivative formula based approach for a shape optimization problem with constrained coupled problems
Azeddine SADIK, Abdesslam BOULKHEMAIR, Abdelkrim CHAKIB In this paper, we deal with numerical method for the approximation of a class of coupled shape optimization problem, which consist in minimizing an appropriate general cost functional subjected to coupled boundary value problems by means of a Neumann boundary transmission condition. We show the existence of the shape derivative of the cost functional and express it by means of support functions. Then the numerical discretization is performed using the dual reciprocity boundary element method. Finally, we give some numerical results,showing the efficiency of the proposed approach. 
Piecewise Linear Bounding of the Euclidean Norm
Aloïs Duguet, Christian Artigues, Laurent houssin, Sandra Ulrich Ngueveu In the field of piecewise linear approximation of nonlinear functions, few studies focused on the minimization of the number of pieces for a given error bound. In this talk we focus on the euclidean norm defined on a plane, and we first show how to use scalar products with regularly spaced unit vectors in order to compute piecewise linear approximations with the minimum number of pieces for a given relative error bound. Then we extend the approach to norms with elliptic level sets. An application to the beam layout problem validates the tractability of the method.
TE05: Optimal Control and Optimization in Economics, Finance and Management II
Stream: Optimal Control and Optimization in Economics, Finance and Management
Room: Pontryagin
Chair(s):
GerhardWilhelm Weber

Risk measure and portfolio selection
BEN HSSAIN LHOUCINE, Ghizlane Lakhnati This paper deals with risk measurement and portfolio optimization under risk constraints. Firstly, we give an overview of Extended Gini shortfall (EGS), a special type of Spectral Measures of Risk that extends the Ginitype measures of risk and variability. We subsequently considering the portfolio problems based on Extended Gini shortfall and gets some useful results. 
Investigating the reliability of the RBF method for solving some optimal control problems in Dynamic Investment
Ahmad Saeedi, Ahamd golbabai In this paper we have developed a MATLAB code based on Radial Basis Function (RBFs) for solving optimal control problems arising in Dynamic Economic Models. Some operational matrices has been produced which helps to reduce the computational cost. The proposed method has good accuracy but it suffers from illconditioning. In order to produce more reliable solutions, the effective condition number and some strategies including BiCGSTAB and GMRES iterative solution methods have been used. 
Market making policy for highfrequency trading
Francis HuotChantal, Fabian Bastin, Gabriel Yergeau We describe a quoting policy for market making based on inventory models optimization, leading to a threshold policy, in the context of highfrequency trading. In our model, the market maker aims to maximize his profit while providing liquidity to the stock market and keeping the risk at a reasonable level. We assess the policy performance in terms of profitability and robustness using backtesting and agentbased simulation. Various simulation frameworks are considered, using policies based on time intervals or discrete events, and a zerointelligence policy as a baseline.
TE06: Optimization under Uncertainty and Applications II
Stream: Optimization under Uncertainty and Applications
Room: Moreau
Chair(s):
Riccardo Cambini

An integrated multiechelon robust closed loop supply chain under imperfect quality production
Theodore Trafalis, Ismail Almaraj In this paper, we consider a novel closed loop supply chain design consisting of multiple periods and multiple echelons. The models are considered under imperfect quality production with multiple uncertainties to provide meaningful solutions to practical problems. In addition, we assume that the screening is not always perfect, and inspection errors are more likely to take place in practice. We measure the amount of quality loss as conforming products deviate from the specification (target) value. In our study, we develop three robust counterparts models based on box, polyhedral, and combined. 
Robust Twostage Polynomial Optimization
Bissan Ghaddar, Olga Kuryatnikova, Daniel Molzahn In this work we consider twostage polynomial optimization problems under uncertainty. We combine tools from polynomial and robust optimization to provide a framework for general adjustable robust polynomial optimization problems. In particular we propose an iterative algorithm to build a sequence of (approximately) robustly feasible solutions with an improving objective value and verify robust feasibility or infeasibility of the resulting solution under a semialgebraic uncertainty set. We implement our approach for a usecase in energy systems to show the performance of the proposed approach. 
Hierarchical Fleet Mix Problems with riskaversion: a CVaR approach
Riccardo Cambini, Rossana Riccardi In this paper a twostage stochastic hierarchical workforce model is studied from both a theoretical and an algorithmic point of view. In the considered hierarchical model workforce units can be substituted by higher qualified ones; external workforce can also be hired to cover unfulfilled jobs. Demand for jobs is assumed to be stochastic. The results of a computational test are provided in order to validate the model.
Friday
Friday, 9:00 – 10:40
FA01: Mathematical Analysis of Optimization Methods I
Stream: Mathematical Analysis of Optimization Methods
Room: Fermat
Chair(s):
David Müller

Decompositions and projections with respect to some classes of nonsymmetric cones
JeinShan Chen In this talk, we focus on issues of decompositions and projections w.r.t. some classes of nonsymmetric cones. These concepts and the obtained results pave a way to deal with nonsymmetric cone optimization. 
Strong duality, boundedness and a theorem of the alternatives in conic optimization
Akshay Gupte Besides strict feasibility, there are other sufficient conditions for strong duality in conic optimization that are somewhat less wellknown, these being a closedness condition and boundedness of feasible region inside a pointed cone. We generalize the closedness condition and show that the other two follow as a consequence of it. We also give an alternate proof for the result that if one problem is strictly feasible then it has finite optimum if and only if the other problem is feasible. Finally, we also give a theorem of the alternative that generalizes what is known for proper cones. 
Properties of Generalized Oriented Distance Function and its Applications to Set Optimization Problems
Pradeep Kumar Sharma In this talk, we discuss several interesting properties of generalized oriented distance function with respect to coradiant sets and free disposal sets, which are more general than a cone. In particular, we deal with some special properties, like, translativity, subadditivity and monotonicity properties using coradiant sets so that this function can be used as a coherent measure of investment in financial mathematics. As an application, we study some optimality conditions for quasiminimal solutions for set optimization problems. 
Discrete choice proxfunctions on the simplex
David Müller We derive new proxfunctions on the simplex from additive random utility models of discrete choice. They are convex conjugates of the corresponding surplus functions. In particular, we explicitly derive the convexity parameter of discrete choice proxfunctions associated with generalized extreme value models and specifically with generalized nested logit models. Incorporated into subgradient schemes, discrete choice proxfunctions lead to natural probabilistic interpretations of the iteration steps. We also discuss an economic application of discrete choice proxfunctions in consumer theory.
FA02: Beyond Firstorder Optimization Methods for Machine Learning – Part III
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Fred Roosta, Albert Berahas

Largescale derivativefree optimization using subspace methods
Lindon Roberts, Coralia Cartis In existing techniques for modelbased derivativefree optimization, the computational cost of constructing local models and Lagrange polynomials can be high. As a result, these algorithms are not as suitable for largescale problems as derivativebased methods. In this talk, I will discuss a modelbased derivativefree algorithm based on exploration of random subspaces, its worstcase complexity bounds, and some numerical results. 
Efficient Newton Methods for Robust Stochastic Nonconvex Optimization
Thomas O’LearyRoseberry, Nick Alger, Omar Ghattas In this talk we explore how low dimensional geometric information of Hessians arising in stochastic nonconvex optimization problems can be used to design matrixfree Newton methods that have similar periteration computational complexity to first order methods. These methods can reduce the dependence on hyperparameters and improve convergence. Further these methods can be adapted to guard against overfitting by using the Hessian to detect noise dominated geometric information of the stochastic energy landscape. 
A stochastic second ordertype extrastep scheme for nonsmooth nonconvex optimization
Andre Milzarek, Minghan Yang, Zaiwen Wen, Tong Zhang We present a stochastic second ordertype extrastep method for solving nonsmooth nonconvex optimization problems. We assume that gradient and Hessian information of the smooth part of the objective function can only be approximated. Our method combines stochastic higher order steps for an underlying optimality condition with stochastic proximal steps to ensure convergence in expectation. A variant of our approach using variance reduction is discussed and we show that it enjoys better convergence properties. Numerical results on logistic regression and sparse deep learning are provided. 
Doubly Adaptive Scaled Algorithm for Machine Learning Using SecondOrder Information
Martin Takac, Majid Jahani, Sergey Rusakov, Zheng Shi, Peter Richtarik, Michael Mahoney This paper presents a novel adaptive optimization algorithm for largescale machine learning problems. Equipped with a lowcost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and stepsize. The search direction contains the gradient information wellscaled by a diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter.
FA03: Equilibria, variational models and applications
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s):
Mauro Passacantando

MultiLeaderFollower Potential Games
Steffensen Sonja We discuss a particular class of Nash games, where the participants of the game (the players) are devided into to groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996. We develop necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results. 
A Decomposition Algorithm for Nash Equilibria in Intersection Management
Andreas Britzelmeier, Axel Dreves We consider autonomous vehicles that communicate in order to find optimal and collision free driving strategies. Resulting in coupled optimal control problems with nonconvex shared constraints and we consider a generalized Nash equilibrium reformulation of the problem. To handle the nonconvexity, we employ a partial penalty approach and reformulate the problem as a generalized potential game. We propose a decomposition algorithm with penalty selection to avoid a priori hierarchies. Using dynamic programming, we prove convergence of our algorithm. Providing numerical and experimental results. 
Lagrange multipliers for a nonconstant gradient constraint problem
SOFIA GIUFFRE’ Aim of the talk is to show the existence of Lagrange multipliers associated with linear variational inequalities with a nonconstant gradient constraint over convex domains. In particular, first we show the equivalence between the nonconstant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a HamiltonJacobi equation in the viscosity sense. Then, we obtain the existence of Lagrange multipliers associated to the problem. The results have been obtained, using the strong duality theory. 
Radial solutions to a nonconstant gradient constraint problem
Attilio Marcianò, SOFIA GIUFFRE’ Our work deals of radial solutions to a nonconstant gradient constraint problem in a ball of R^n and the Lagrange multipliers associated with the problem. The problem is formulated by means of a variational inequality and under a suitable assumption we obtain, for n = 2, a necessary and sufficient condition, characterizing the free boundary. In particular, this condition allow us to define the explicit solution and the Lagrange multiplier. Finally, some examples illustrate the results.
FA04: Multiobjective Mixed Integer Optimization
Stream: Multiobjective Optimization
Room: Lagrange
Chair(s):
Marianna De Santis, Gabriele Eichfelder

An objective space based algorithm for biobjective mixed integer programming problems
Firdevs Ulus, Ozlem Karsu, Deniz Emre We propose an objective space based exact solution algorithm for biobjective mixed integer programming problems. The algorithm solves scalarization models in order to explore predetermined regions of the objective space called boxes, defined by two nondominated points. At each iteration of the algorithm, a box is explored either by a weighted sum or a PascolettiSerafini scalarization to determine nondominated line segments and points. We demonstrate the applicability of the algorithm through computational experiments. 
Toward a BranchandBound Algorithm for Biobjective Mixed Integer Quadratic Programs
Margaret Wiecek Biobjective quadratic programs with mixedinteger variables (BOMIQPs) model decision making situations encountered in finance, economics, engineering, and other areas of human activity. We present the work we have accomplished so far toward development of a branch and bound (BB) algorithm for convex BOMIQPs. The algorithm consists of the following components: (i) algorithms for solving auxiliary single objective parametric quadratic programs; (ii) fathoming rules; (iii) branching rules; (iv) rules to establish (non)dominance between sets; (v) rules for updating the incumbent nondominated set. 
Finding nondominated and efficient solutions of multiobjective integer quadratic programming problems
Marianna De Santis, Gabriele Eichfelder We present a deterministic method for minimizing multiple quadratic objective functions over integer variables. Our method looks for efficient points by fixing subsets of variables to integer values and by using lower bounds in the form of hyperplanes in the image space derived from the continuous relaxations of the restricted objective functions. We show that the algorithm finds all efficient and all nondominated points after finitely many fixings of variables. Numerical examples on biobjective instances as well as on instances with three and four objectives are shown. 
Pareto Front Approximation through a Multiobjective Augmented Lagrangian Method
Pierluigi Mansueto, Matteo Lapucci, Guido Cocchi In this work, we propose an extension of a multiobjective augmented Lagrangian Method from recent literature suitable for smooth convex constrained multiobjective optimization problems. The new algorithm is designed to handle sets of points and produce good Pareto front approximations, as opposed to the original one which converges to a single solution. We prove properties of Pareto stationarity global convergence for the sequences of points generated by our method. We then compare it with main stateoftheart algorithms on some problems, showing its effectiveness and general superiority.
FA05: Optimization and Artificial Intelligence II
Stream: Optimization and Artificial Intelligence
Room: Pontryagin
Chair(s):
Sébastien Gerchinovitz

KQuant: a non uniform posttraining quantization algorithm
Enrico Civitelli, Leonardo Taccari, Fabio Schoen Quantization is a simple yet effective way to deploy deep neural networks on resourcelimited hardware. Posttraining quantization algorithms are particularly interesting because they do not require the full dataset to run. In this work we explore a way to perform non uniform posttraining quantization using an optimization algorithm to minimize the output differences between each compressed layer and the original one. The proposed method significantly reduces the memory required by the neural network without affecting the performance in terms of accuracy. 
An Adaptive MLBased Discretization Method for Computing Optimal Experimental Designs
Philipp Seufert, Jan Schwientek, Tobias Seidel, Michael Bortz, KarlHeinz Küfer Standard algorithms for the computation of optimal experimental designs (OED) consist of an inner point acquisition and an outer weight optimization. Whereas the latter is a convex problem, the inner one is a general nonconvex nonlinear program with implicitly given objective. We present a modification of the common OED solution approach which uses Bayesian optimization to adaptively form a grid of candidate points for determining the optimal design. We proved convergence of the algorithm to a locally optimal continuous design and obtained promising numerical results on realworld problems. 
Sparse RBF Regression for the Optimization of Noisy Expensive Functions
Alessio Sortino, Matteo Lapucci, Fabio Schoen Global optimization problems for blackbox functions are usually addressed by building a surrogate model over the data and an acquisition function to decide where to place the next observation. When data are noisy the surrogate should not trust the latter too much. This typically introduces an extra hyperparameter into the model that corresponds to the variance of the noise. In this work we present a novel approach where a robust RBFbased surrogate model is built from the solution of a particular MIQP problem. Experimental results show the effectiveness of our approach w.r.t. existent methods 
On the optimality of the PiyavskiiShubert algorithm for global Lipschitz optimization: a bandit perspective
Clément Bouttier, Sébastien Gerchinovitz, Tommaso Cesari We consider the problem of maximizing a nonconcave Lipschitz multivariate function over a compact domain by sequentially querying its (possibly perturbed) values. We study a natural algorithm originally designed by Piyavskii and Shubert in 1972, for which we prove new bounds on the number of evaluations of the function needed to reach or certify a given optimization accuracy. Our analysis uses a banditoptimization viewpoint and solves an open problem from Hansen et al. (1991), by bounding the number of evaluations to certify a given accuracy with a simple and optimal integral.
FA06: DerivativeFree Optimization II
Stream: DerivativeFree Optimization
Room: Moreau
Chair(s):
Clément Royer

Distributed DerivativeFree Optimization
Vyacheslav Kungurtsev There are a number of situations in which multiple agents are seeking to cooperatively solve a central optimization problem using a combination of computations using local information and communication across a network modeled as a graph. In contexts arising from hyperparameter tuning or the cooperative control of functions defined by simulations, these problems are derivative free. In this talk we study the classical DFO algorithms adapted for distributed optimization. The theoretical properties and numerical performance is presented and assessed. 
Derivativefree optimization for largescale structured problems
Andrea Cristofari, Francesco Rinaldi In this talk, a derivativefree algorithm is proposed to minimize a blackbox objective function over the convex hull of a given set of points. At each iteration, the approach suitably handles a reduced problem, based on an inner approximation of the feasible set. This inner approximation is dynamically updated by using rules that in general allow us to keep the dimension of the problem small. Theoretical convergence results and numerical comparison with stateoftheart DFO solvers will be presented. 
A stochastic LevenbergMarquardt method using random models
Clément Royer, El houcine Bergou, Youssef Diouane, Vyacheslav Kungurtsev In this talk, we describe a stochastic LevenbergMarquardt algorithm that handles noisy objective function values and random models. Provided the probability of accurate function estimates and models is sufficiently large, we are able to endow our proposed framework with global convergence rates. Our results rely on both a specific scaling of the regularization parameter and a measure of criticality tailored to leastsquares problems. We compare our approach with several rates available for stochastic algorithms, and discuss the assumptions under which these schemes can be analyzed. 
A Derivativefree Adaptation of the Penalty Decomposition Method for Sparse Optimization
Matteo Lapucci, Marco Sciandrone We consider the problem of minimizing a smooth function with cardinality constraint. A wellknown approach to tackle such prolem is the Penalty Decomposition (PD), which requires to exactly solve subproblems in the dimension of the original problem. In this talk, we present a derivativefree PD algorithm, where the exact minimization step is replaced by inexact line searches along suitable sets of directions. We state theoretical convergence properties of the proposed algorithm equivalent to those of the original gradientbased PD method. Finally, we show the results of preliminary experiments
Friday, 11:00 – 12:40
FB01: Mathematical Analysis of Optimization Methods II
Stream: Mathematical Analysis of Optimization Methods
Room: Fermat
Chair(s):
JeanBaptiste HiriartUrruty

The simplest KKT and FJ optimality conditions via subdifferential calculus of pointwise suprema
Abderrahim Hantoute We present first some new characterizations of the subdifferential of pointwise suprema. Compared to the previous results in the literature, the present ones are given only by means of the data functions. So, no need to consider the normal cone to the ddomain of the supremum nor to require intersections over finitedimensional subspaces. Based on this, we develop some new KKT and FJ optimality conditions for a general convex optimization problem with (possibly) infinitely many constraints. The results discussed here are base on a recent work joint with M. A. López and R. Correa. 
Optimality conditions for an exhausterable function on an exhausterable set
Majid Abbasov Exhausters were proposed by V.F.Demyanov. These are families of convex compact sets that allow one to represent the directional derivative of the studied function at a point in the form of minmax or maxmin of linear functions. Functions for which such a representation is valid we call exhausterable. The set which is defined via exhausterable function is also called exhausterable. In the present work we describe optimality conditions for an exhausterable function on an exhausterable set. The reported study was supported by Russian Science Foundation (RSF), project No. 207110032 
Minimal sublinear functions and application to Cut Generating Functions
alberto zaffaroni Given a convex set V, its recession hull consists in taking the intersection of all translates of the recession cone which contain V. We first study its main features and characterizations. Based on this, we study minimality of sublinear functions, among those for which the 1lower level set has a prescribed recession cone. We prove that if F is recession minimal, then its lower 1level set is regular, in the sense that it coincides with its recession hull. At last we apply the results above to Cut Generating Functions, and provide a complete characterization of minimal CGFs. 
A fresh geometrical look at the general Sprocedure
JeanBaptiste HiriartUrruty , Michel De Lara We revisit the Sprocedure for general functions with “geometrical glasses”. We thus delineate a necessary condition, and almost a sufficient one, to have the Sprocedure valid. Everything is expressed in terms of convexity of augmented sets (convex hulls, conical hulls) of images built from the data functions.
FB02: Advances in DouglasRachford method – Part II
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Cong Bang Vu, Dimitri Papadimitriou

Shadow DouglasRachford splitting
Matthew Tam In this talk, I will introduce the shadow DouglasRachford method for finding a zero in the sum of two monotone operators where one is assumed to be singlevalued and Lipschitz continuous. This algorithm naturally arises from a nonstandard discretisation of a continuous dynamical system associated with the Douglas–Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretisation with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator. 
The Cyclic DouglasRachford Algorithm with rsetsDouglasRachford Operators
Aviv Gibali, Francisco Javier Aragón Artacho, Yair Censor The DouglasRachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this talk we present a structural generalization that allows to use rsetsDR operators in a cyclic fashion and also demonstrates great computational advantages over existing results. 
Forwardpartial inversehalfforward splitting algorithm for solving monotone inclusions with applications
Yuchao Tang In this paper, we propose a forwardpartial inversehalfforward splitting (FPIHFS) algorithm for finding a zero of the sum of a maximally monotone operator, a monotone Lipschitzian operator, a cocoercive operator, and a normal cone of a closed vector subspace. The FPIHFS algorithm is derived from a combination of the partial inverse method with the forwardbackwardhalfforward splitting algorithm. As applications, we apply it to solve the Projection on Minkowski sums of convex sets problem and the generalized Heron problem. 
Multivariate Monotone Inclusions in Saddle Form
Patrick Combettes, Minh Bui We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. We study a highly structured multivariate monotone inclusion problem involving a mix of setvalued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicitypreserving operations among them. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, uses the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to all of them. Applications are presented.
FB03: Game theory, multilevel and dynamic optimization II
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s):
Giancarlo Bigi

NearOptimal Robust Bilevel Optimization
Miguel F. Anjos, Mathieu Besançon, Luce Brotcorne We introduce nearoptimal robustness for bilevel optimization problems to protect the upperlevel decisionmaker from bounded rationality at the lower level. We show that it is a restriction of the corresponding pessimistic bilevel problem. Essential properties are derived in generic and specific settings. This model has an intuitive interpretation in various situations cast as bilevel optimization problems. We develop a dualitybased solution method for cases where the lower level is convex. The models obtained are successfully tested numerically using different solvers and formulations. 
A Bilevel Optimization Model for Generative Adversarial Networks
Zeynep Suvak, Miguel F. Anjos, Luce Brotcorne, Diego Cattaruzza A generative adversarial network is a framework to model an adversarial process where a generator produces data and a discriminator decides whether the data is real or fake. We focus on the example where the generator is a counterfeiter producing fake banknotes and the discriminator is the police trying to detect the fakes. This problem is formulated as a bilevel optimization problem where the counterfeiter determines the banknotes to be given to the police and the police decides which features to use to check authenticity. The details of the model and computational results will be presented. 
Best response approaches for optimization problems with equilibrium constraints
Maria Carmela Ceparano, Francesco Caruso, Jacqueline Morgan We consider a oneleader twofollower Stackelberg game where the leader’s payoff depends on the followers’ actions but the followers’ payoffs do not depend on the leader’s actions. We first present a theoretical method exploiting affine relaxations of the classical best response algorithm which globally converges to an equilibrium for a class of Stackelberg games where the uniqueness of the Nash equilibrium of the game played by the followers is ensured. Then, the convergence of a derivativefree numerical method will be shown, together with error bounds. 
An explicit Tikhonov algorithm for nested variational inequalities
Lorenzo Lampariello, Christoph Neumann, Jacopo Maria Ricci, Simone Sagratella, Oliver Stein We consider nested variational inequalities consisting in a (upperlevel) variational inequality whose feasible set is given by the solution set of another (lowerlevel) variational inequality. Purely hierarchical convex bilevel optimization problems and certain multifollower games are particular instances of nested variational inequalities. We present an explicit and readytoimplement Tikhonovtype solution method for such problems. We give conditions that guarantee global strong convergence of the proposed method and we provide a convergence rate analysis.
FB04: Numerical Methods for Multiobjective Optimization
Stream: Multiobjective Optimization
Room: Lagrange
Chair(s):
Leo Warnow, Gabriele Eichfelder

An efficient descent method for locally Lipschitz multiobjective optimization problems
Bennet Gebken In this talk, we propose a new descent method for MOPs with locally Lipschitz objective functions and afterwards combine it with the subdivision method to compute the whole Pareto set. The descent direction is based on epsilonsubdifferentials, which are iteratively enriched with gradient information until a satisfying direction is found. Combined with a modified Armijo step length, we prove convergence of the method to points that satisfy a necessary condition for Pareto optimality. Finally, the descent method is inserted into the subdivision method to compute the whole Pareto set. 
Worstcase complexity bounds of directional directsearch methods for multiobjective derivativefree optimization
Rohollah Garmanjani, Ana Luisa Custodio, Youssef Diouane, Elisa Riccietti Direct Multisearch (DMS) is a wellestablished class of algorithms for multiobjective derivativefree optimization. We analyze the worstcase complexity of this class of methods in its most general form, for the class of nonconvex smooth functions. We then focus on a particular instance of DMS, which considers a stricter criterion for accepting new nondominated points and has a worstcase complexity bound similar to the one obtained for gradient descent for the same class of problems (of the order of a threshold raised to the power 2, for driving a criticality measure below this threshold). 
A normminimization based convex vector optimization algorithm
Muhammad Umer, Firdevs Ulus, Cagin Ararat We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm minimizing scalarizations, which differ from PascolletiSerafini scalarization in a sense that it does not involve a direction parameter. Therefore, the algorithm is free from direction biasedness. The finiteness of the algorithm is shown, and the convergence rate of the algorithm is studied under additional assumptions. 
Computing a boxcoverage for multiobjective optimization problems
Leo Warnow, Gabriele Eichfelder In this talk, we present how to compute a boxcoverage of the nondominated set of a continuous multiobjective optimization problem. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. We also present some key results for that algorithm. For example we show that we can guarantee an improvement of the boxes in each iteration and that the algorithm terminates in finite time with a finite number of boxes.
FB05: Optimal Control and Applications II
Stream: Optimal Control and Applications
Room: Pontryagin
Chair(s):
Björn Martens

A new approach for solving the multidimensional control optimization problems with firstorder PDEs constraints
Anurag Jayswal, Tadeusz Antczak, Preeti Kardam This paper aims to solve the multidimensional control optimization problem involving firstorder PDEs constraints (MCOP). Firstly, we apply the modified objective function approach to (MCOP) and show that the solution sets of the original problem and its associated modified problem are equivalent. Further, we apply the penalty function method to transform the modified problem into an equivalent penalized problem. We also establish the relationship between a saddlepoint of the modified problem and a minimizer of its penalized problem. We also give examples to verify the establish results. 
Multimaterial topology optimization of a heated channel
Alexandre Vieira The presentation will deal with a topology optimization problem in a heated channel. The problem, modelled using the incompressible NavierStokes equations coupled with the convectiondiffusion equation, consists in minimizing pressure drop and maximizing heat transfer for some application in reduced mechanical air conditioning. The analysis focuses on the existence of a solution, the convergence of the numerical approximation using finite elements (including the convergence of the approximated optimum, using controls in BV), and the effect of using multiple materials to design the topology. 
On Preconditioners for PDEConstrained Optimization Problems with HigherOrder Discretization in the Time Variable
Santolo Leveque, John Pearson Optimization problems with PDE constraints arise in numerous scientific applications. In this talk, we consider preconditioners for timedependent PDE optimization, suitably discretized in time. The stateoftheart is to apply a backward Euler method: this leads to desirable structures within the resulting matrix system, and effective preconditioners, but only a firstorder accurate method. Here we present a secondorder method, using CrankNicolson in time, and a new preconditioner for the more complex matrix. We show that this approach can obtain a more accurate solution in less CPU time. 
Convergence
analysis for approximations of optimal control problems subject to
higher index differentialalgebraic equations and mixed controlstate
constraints
Björn Martens, Matthias Gerdts This paper establishes a convergence result for implicit Euler discretizations of optimal control problems with DAEs of higher index and mixed controlstate constraints. The main difficulty of the analysis is caused by a structural discrepancy between the necessary conditions of the continuous problem and the necessary conditions of the discretized problems. This discrepancy does not allow one to compare the respective necessary conditions directly. We use an equivalent reformulation of the discretized problems to overcome this discrepancy and to prove first order convergence.
FB06: Advanced Optimization Methods II
Stream: Advanced Optimization Methods
Room: Moreau
Chair(s):
Jacek Gondzio

New Bregman proximal type algorithms for solving DC optimization problems
Shota Takahashi, Mituhiro Fukuda, Mirai Tanaka Difference of convex (DC) optimization problems have objective functions that are differences of two convex functions. A proximal DC algorithm (PDCA) solves largescale DC optimization problems, assuming the Lsmoothness of objective functions. In this talk, using the Bregman distances, we propose a Bregman proximal DC algorithm (BPDCA) that does not require the Lsmoothness. In addition, accelerating the BPDCA by the extrapolation technique, we propose the Bregman proximal DC algorithm with extrapolation (BPDCAe). 
Parameter tuning of linear programming solvers
Nikolaos Ploskas Linear programming solvers have a large set of parameters that allow users to control algorithmic aspects. Tuning solver options may have a considerable impact on solver performance. Previous efforts to tune solver parameters have used derivativefree optimization algorithms. In this work, we apply various derivativefree optimization solvers in order to find high quality tuning parameters for linear programming solvers. This work investigates how sensitive linear programming solvers are to a parameter tuning process. We present extensive computational results. 
A modified barrier method with algebraic multigrid solver
Alexander Brune, Michal Kocvara The PenaltyBarrier Multiplier (PBM) method, originally introduced by R. Polyak and later studied by BenTal and Zibulevsky and others proved to be an efficient tool for solving very large scale optimization problems. The numerical bottleneck of the method (as of similar methods) lies in the repeated solution of a very large linear systems. We will show that for problems resulting from topology optimization of mechanical structures on irregular finite element meshes, algebraic multigrid can be used to solve problems of large dimension, unsolvable by direct methods. 
A new stopping criterion for PCG applied in IPM
Filippo Zanetti, Jacek Gondzio When Conjugate Gradient Method is used as a linear solver in the Interior Point Method, the attention is usually placed on accelerating its convergence by designing appropriate preconditioners, and the PCG is applied as a black box solver. We design and analyze a new specialized termination criterion for PCG applied in the context of IPMs. The new criterion has been tested on a set of linear and quadratic optimization problems including compressed sensing and image processing and has demonstrated consistent and significant improvements in terms of CG iterations and computational time.
Friday, 14:00 – 15:00
FC01: Plenary 2 – Oliver STEIN
Stream: Plenary
Room: Fermat
Chair(s):
Giancarlo Bigi

Granularity — a bridge between continuous and discrete optimization
Oliver Stein In this talk we sketch the development of the granularity concept in optimization over the past five years. Granularity relaxes the difficulties imposed by integrality conditions and often provides ways for determining good feasible points of mixedinteger optimization problems at low computational cost. Starting from error bound results for roundings in mixedinteger linear optimization, we illustrate how this concept unfolded to provide algorithms for the computation of feasible points in mixedinteger linear, convex and nonconvex optimization. We also comment on the treatment of equality constraints and explain the integration of the granularity idea into branchandbound frameworks.
Friday, 15:15 – 16:30
FD01: DerivativeFree Optimization III
Stream: DerivativeFree Optimization
Room: Fermat
Chair(s):
Youssef Diouane

Optimization of noisy blackboxes with adaptive precision
PierreYves Bouchet, Charles Audet, Sébastien Le Digabel, Stephane Alarie In derivativefree optimization, the objective may be evaluated through a computer program which returns a stochastic output with tunable variance. As a low variance leads to an important cost per evaluation, we propose an approach which allows the variance to remain high. Some satisfying results on an industrial problem are given. 
A derivative free methods for mixedinteger nonsmooth constrained optimization problems.
Stefano Lucidi, Tommaso Giovannelli, Giampaolo Liuzzi, Francesco Rinaldi In this paper, we propose new linesearchbased methods for mixedinterger nonsmooth constrained optimization problems when firstorder information on the problem functions is not available. First, we describe a general framework for mixed integer bound constrained problems. Then we use an exact penalty approach to tackle the presence of nonlinear (possibly nonsmooth) constraints. We analyze the global convergence properties of all the proposed algorithms toward Clarke stationary points and we report results of a preliminary numerical experience on a set of mixedinteger problems. 
A Merit Function Approach for Evolution Strategies
Youssef Diouane In this talk, we present a class of globally convergent evolution strategies for solving relaxable constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The introduced extension guarantees to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Comparison tests are carried out using two sets of known constrained optimization problems.
FD02: Advances in mathematical optimization for machine learning and data analysis – Part III
Stream: Advances in mathematical optimization for machine learning
Room: Turing
Chair(s):
Dimitri Papadimitriou

Mixed integer optimization in ARMA models
Leonardo Di Gangi Model selection and fitting represent two critical aspects of Auto Regressive Moving Average (ARMA) models. It is proposed an algorithm which performs the selection and estimation of ARMA models as a single optimization routine without any statistical knowledge. The computational core of the algorithm is based on a twostep GaussSeidel decomposition scheme, where at each iteration the first step involves the update of the autoregressive and moving average parameters by solving a Mixed Integer Optimization problem and the second step the closed form update of the variance parameter. 
Model of Optimal Centroids Approach For Multivariate Data Classification
Pham Van Nha, Le Cam Binh Particle swarm optimizationPSO is a wellknown multidisciplinary optimization algorithm. However, the general mathematical model of PSO has not been presented. In this paper, PSO will be presented as a general mathematical model and applied in multivariate data classification. First, PSO’s the general mathematical model is analyzed so that can be applied into complex applications. Then, Model of Optimal CentroidsMOC is proposed for multivariate data classification. Experiments were conducted on some data sets to demonstrate the effectiveness of MOC compared with some proposed algorithms 
IGLOO: A stochastic global optimization algorithm to predict the structure of biomolecules adsorbed on metal surfaces
Juan CORTES, Nathalie TARRAT, Christian Schoen Predicting conformations of molecular systems constitutes a global optimization problem for an appropriate cost (energy) function, where not only the global minimum but also a representative set of local minima is expected as output. Depending on the system size and the complexity of the energy function, solving this optimization problem can be extremely challenging. Our proposed Iterative Global exploration and LOcal Optimization (IGLOO) algorithm iterates RRTbased sampling, local minimization and clustering, until convergence is achieved. We show its performance in a real application.
FD03: Advances in Operator Splitting Techniques
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s):
Mathias Staudigl

On optimal gradient methods and oracle complexity for smooth (possibly strongly) convex minimization
Adrien Taylor In this talk, we want to present the (definitely) optimal (blackbox) gradient method for smooth strongly convex minimization, which we called the InformationTheoretic Exact Method (ITEM). This method was obtained through semidefinite programming, and we humbly believe it provides some nice insights on the topic of accelerated methods. On the way, we will discuss the corresponding (matching) lower complexity bounds and a constructive procedure for obtaining them, as well as a few links between accelerated and conjugate gradient methods. This talk is based on joint works with Yoel Drori. 
The sample complexity of level set estimation
François Bachoc, Tommaso Cesari, Sébastien Gerchinovitz The problem of approximating the level set of a real function f of many real variables arises naturally in many fields, including physics, engineering, and computer science. We investigate the minimum number of evaluations of f needed to output a guaranteed approximation of such level set. We study the general case of Lipschitz functions f as well as other, smaller function spaces. In all the cases we consider, we show that such sample complexity is exponential in the number of variables. Additionally, we provide algorithms and intuitions on how to attain optimal sample complexity rates. 
Multiblock Bregman proximal alternating linearized minimization for structured nonsmooth nonconvex problems
Masoud Ahookhosh, Le Thi Khanh Hien, Nicolas Gillis, Panagiotis Patrinos We introduce BPALM and ABPALM, two multiblock proximal alternating linearized minimization algorithms using Bregman distances for the sum of a multiblock relatively smooth function and block separable (nonsmooth) nonconvex functions. Our algorithms are globally convergent to critical points of the objective function under KL assumption, and their rate of convergence is studied for Lojasiewicztype KL functions. We apply this framework to orthogonal nonnegative matrix factorization (ONMF) that satisfies all of our assumptions and the related subproblems are solved in closed forms.
FD05: Optimal Control and Optimization in Economics, Finance and Management III
Stream: Optimal Control and Optimization in Economics, Finance and Management
Room: Pontryagin
Chair(s):
Diogo Pinheiro

Stochastic Maximum Principle with Regimes and Memory
Emel Savku, GerhardWilhelm Weber We study a stochastic optimal control problem for a delayed Markov regime switching jumpdiffusion model. We establish necessary and sufficient maximum principles for such a system.We prove the existence– uniqueness theorem for the adjoint equations, which are represented by an anticipated backward stochastic differential equation with jumps and regimes. We illustrate our results by a problem of optimal consumption problem from a cash flow with delay and regimes. 
Portfolio Optimization with Drift Uncertainty
Kerem Ugurlu We study the utility maximization problem of a portfolio of one risky asset, a stock, and one riskless asset, a bond, under Knightian uncertainty on the drift term representing the long term growth rate of the risky asset. We further assume that the investor has a prior estimate about the drift term, so that we incorporate into the model a penalty term for deviating from the prior about the mean. We provide explicit solutions, when the investor has logarithmic, power and exponential utility functions. 
Twoplayer zerosum stochastic differential games with Markovswitching jumpdiffusion dynamics
Diogo Pinheiro, Miguel Ferreira, Susana Pinheiro We consider a twoplayer zerosum stochastic differential game with Markovswitching jumpdiffusion state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial integrodifferential equation of HamiltonJacobiBellmanIsaacs type.
FD06: Applications of Optimization IV
Stream: Applications of Optimization
Room: Moreau
Chair(s):
Gabriella Colajanni

Volumetric Uncertainty Bounds and Optimal Configurations for Converging Beam Triple LIDAR
Anthony Brooms, Theodore Holtom We consider the problem of forward uncertainty propagation for the converging beam triple LIDAR technology, used for measuring wind velocity passing through a fixed point in space. The size of the volumetric output uncertainty is related to the inverse of the volume of a parallelepiped of unit edge length, delineated by the Doppler LIDAR configuration. Optimal configurations for minimizing output uncertainty are discussed, whilst a grid search procedure for optimizing the value of the parallelepiped volume, subject to LIDAR orientation uncertainty constraints, is presented. 
Optimal Incentive in Electric Vehicle Adoption
Giorgio Rizzini We study a bilevel model with a policymaker and a population of vehicle owners. The policymaker minimizes a cost function deciding the incentive to encourage the largest possible percentage of the fossilfueled vehicle owners to buy an electric one. All players care about PM10 concentration. The policymaker imposes a traffic ban if the PM10 concentration exceeds a safety threshold. Traffic bans generate a cost to the owners of a fossilfueled vehicle. We reduce the initial bilevel formulation to a single level problem, which is solved analytically. 
An Optimization Model to manage UAVs in multitiered 5G network slices
Daniele Sciacca, Gabriella Colajanni In this paper, we present a threetier supply chain network model consisting of a fleet of UAVs organized as a FANET managed by a fleet of UAVs controllers, whose purpose is to provide 5G network slices on demand to users and devices on the ground. Our aim is to provide a system optimization perspective for the entire supply chain network, obtaing a constrained optimization problem for which we derive the associated variational inequality formulation. Also, qualitative properties in terms of existence and uniqueness of solution are provided.
Friday, 16:45 – 17:15
FE01: Closing
Stream: Closing
Room: Fermat