This talk presents recent work on solving multivariate polynomial optimization problems by using semidefinite programming (SDP) and sum of squares (SOS) techniques. It consists of two parts. The first part focus on Lasserre type SDP relaxation: sum of squares polynomials, Lasserre's relaxation hierachy, its convergence, and approximation peformance analysis. The second part focus on Jacobian type SDP relaxation: gradient SOS methods, Jacobian SOS methods, minimum defining equations of determinantal varieties,and the exactness of Jacobian type relaxation in solving polynomial optimization.
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