Generic Minimizing Behavior in Semialgebraic Optimization

Abstract

We present a theorem of Sard type for semialgebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, for a generic value parameter. This simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems: “typical” semialgebraic problems have finitely many critical points, around each of which they admit a unique “active manifold” (analogue of an active set in nonlinear optimization); moreover, such critical points satisfy strict complementarity and second-order sufficient conditions for optimality are indeed necessary.