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arXiv · 1504.07694

Generic minimizing behavior in semi-algebraic optimization

D. Drusvyatskiy·A. D. Ioffe·A. S. Lewis
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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.

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D. Drusvyatskiy, A. D. Ioffe, A. S. Lewis. 2015-04-29. Generic minimizing behavior in semi-algebraic optimization. https://arxiv.org/abs/1504.07694

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