Analysis of Non Linear Algorithms II

WD-01: Analysis of Non Linear Algorithms II
Stream: Analysis of Non Linear algorithms
Room: Fermat
Chair(s): Vladimir Shikhman

An Interior Point-Proximal 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 (IP-PMM) is interpreted as a primal-dual 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 ill-posedness 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 W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints.

Local convergence of tensor methods
Nikita Doikov, Yurii Nesterov
In this talk, we study local convergence of high-order 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 Lipschitz-continuous high-order 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.

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