Survey of Piecewise Convex Maximization and PCMP over Spherical Sets

  • Ider TseveendorjEmail author
  • Dominique Fortin
Part of the Springer Optimization and Its Applications book series (SOIA, volume 107)


The main investigation in this chapter is concerned with a piecewise convex function which can be defined by the pointwise minimum of convex functions, \(F(x) =\min \{ f_{1}(x),\ldots,f_{m}(x)\}\). 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.


Piecewise convex Nonconvex optimization Nonsmooth optimization 



This research benefited from the support of the FMJH “Program Gaspard Monge for optimization and operations research”, and from the support from EDF. The authors acknowledge use of the IBM ILOG CPLEX under the academic initiative license. The authors would like to thank the anonymous referee for his/her useful comments on this chapter.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of VersaillesUniversité Paris-SaclayVersailles CedexFrance
  2. 2.INRIA, Domaine de Voluceau, RocquencourtLe Chesnay CedexFrance

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