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Section: New Results

Global optimization

Let f 1 ,,f p be in [𝐗], where 𝐗=(X 1 ,,X n ) t , that generate a radical ideal and let V be their complex zero-set. Assume that V is smooth and equidimensional. Given f[X] bounded below, consider the optimization problem of computing f =inf xV n f(x). For 𝐀GL n (), we denote by f 𝐀 the polynomial f(𝐀𝐗) and by V 𝐀 the complex zero-set of f 1 𝐀 ,...,f p 𝐀 . In [9] , we construct families of polynomials 𝖬 0 𝐀 ,...,𝖬 d 𝐀 in [𝐗]: each 𝖬 i 𝐀 is related to the section of a linear subspace with the critical locus of a linear projection. We prove that there exists a non-empty Zariski-open set OGL n () such that for all 𝐀OGL n (), f(x) is non-negative for all xV n if, and only if, f 𝐀 can be expressed as a sum of squares of polynomials on the truncated variety generated by the ideal 𝖬 i 𝐀 , for 0id. Hence, we can obtain algebraic certificates for lower bounds on f using semidefinite programs. Some numerical experiments are given. We also discuss how to decrease the number of polynomials in 𝖬 i 𝐀 .