We first show that a continuous function "f" is nonnegative on a closed set K if and only if (countably many) moment matrices of some signed measure dnu = fdmu are all positive semidefinite (if K is compact mu is an arbitrary finite Borel measure with support exactly K). In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with no lifting, of the cone of nonnegative polynomials of degree at most d. Wen used in polynomial optimization on certain simple closed sets K (like e.g., the whole space, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable (in fact, a generalized eigenvalue problem). In the compact case, this convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations already obtained by the author.
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LAAS-CNRS and Institute of Mathematics, Toulouse, France
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View A new look at nonnegativity on closed sets and polynomial optimization