Generic minimizing behavior in semi-algebraic optimization

TL;DR

This paper proves a Sard-type theorem for semi-algebraic set-valued mappings, leading to a unified understanding that typical semi-algebraic optimization problems have finitely many critical points with unique active manifolds and necessary second-order conditions.

Contribution

It introduces a Sard-type theorem for semi-algebraic set-valued mappings, enabling a unified analysis of generic properties in semi-algebraic optimization.

Findings

Semi-algebraic problems have finitely many critical points

Each critical point has a unique active manifold

Second-order conditions are necessary at critical points

Abstract

We present a theorem of Sard type for semi-algebraic 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 semi-algebraic optimization problems: "typical" semi-algebraic 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.