https://hal.uvsq.fr/hal-02173639Tseveendorj, IderIderTseveendorjLMV - Laboratoire de Mathématiques de Versailles - UVSQ - Université de Versailles Saint-Quentin-en-Yvelines - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueFortin, DominiqueDominiqueFortinGANG - Networks, Graphs and Algorithms - LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en AutomatiqueSurvey of Piecewise Convex Maximization and PCMP over Spherical SetsHAL CCSD2016[MATH] Mathematics [math]HAL UVSQ, Équipe2019-07-04 15:50:402022-10-25 16:22:212019-07-04 15:50:40enBook sections10.1007/978-3-319-29975-4_31The main investigation in this chapter is concerned with a piecewise convex function which can be defined by the pointwise minimum of convex functions, F(x)=min{f1(x),…,fm(x)}. Such piecewise convex functions closely approximate nonconvex functions, that seems to us as a natural extension of the piecewise affine approximation from convex analysis. Maximizing F(⋅ ) over a convex domain have been investigated during the last decade by carrying tools based mostly on linearization and affine separation. In this chapter, we present a brief overview of optimality conditions, methods, and some attempts to solve this difficult nonconvex optimization problem. We also review how the line search paradigm leads to a radius search paradigm, in the sense that sphere separation which seems to us more appropriate than the affine separation. Some simple, but illustrative, examples showing the issues in searching for a global solution are given.