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Survey of Piecewise Convex Maximization and PCMP over Spherical Sets

Ider Tseveendorj 1 Dominique Fortin 2 
2 GANG - Networks, Graphs and Algorithms
LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications, Inria Paris-Rocquencourt
Abstract : The 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.
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Ider Tseveendorj, Dominique Fortin. Survey of Piecewise Convex Maximization and PCMP over Spherical Sets. Advances in Stochastic and Deterministic Global Optimization, pp.33-52, 2016, 978-3-319-29975-4. ⟨10.1007/978-3-319-29975-4_3⟩. ⟨hal-02173639⟩



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