Mathematical Analysis of Optimization Methods II

FB-01: Mathematical Analysis of Optimization Methods II
Stream: Mathematical Analysis of Optimization Methods
Room: Fermat
Chair(s): Jean-Baptiste Hiriart-Urruty

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 finite-dimensional 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. 20-71-10032

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 1-lower level set has a prescribed recession cone. We prove that if F is recession minimal, then its lower 1-level 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 S-procedure
Jean-Baptiste Hiriart-Urruty , Michel De Lara
We revisit the S-procedure for general functions with “geometrical glasses”. We thus delineate a necessary condition, and almost a sufficient one, to have the S-procedure valid. Everything is expressed in terms of convexity of augmented sets (convex hulls, conical hulls) of images built from the data functions.

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