Survey of Piecewise Convex Maximization and PCMP over Spherical Sets

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 ) , … , f m ( x ) } $$F(x) =\min \{ f_{1}(x),\ldots,f_{m}(x)\}$$ . 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.