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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.
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth fu
We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a se
We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $mu(om)$ of (not necessarily compact) basic semi-algebraic sets$omsubsetR^n$. We obtain two monotone (non increasing and non decreasing) sequences
We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, tame). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region i
In this paper, we introduce various mechanisms to obtain accelerated first-order stochastic optimization algorithms when the objective function is convex or strongly convex. Specifically, we extend the Catalyst approach originally designed for determ