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In this paper we improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set K ⊂ Rn that is contained in a box of radius R we show how to either compute a point in K or prove that K does not contain a ball of radius ε using an expected O(n log(nR=ε)) evaluations of...
This paper is concerned with algebraic relaxations, based on the concept of sum of squares decomposition, that give sufficient conditions for convexity of polynomials and can be checked efficiently with semidefinite programming. We propose three natural sum of squares relaxations for polynomial convexity based on respectively, the definition of convexity, the first order characterization of convexity,...
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