Generic Minimizing Behavior in Semialgebraic Optimization
本文证明了半代数集值映射的Sard型定理，并应用于优化问题，表明典型半代数问题只有有限个临界点，每个临界点附近存在唯一的活动流形，且满足严格互补性和二阶最优性条件。
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.