Detecting optimality and extracting optimal solutions in polynomial optimization based on the Lasserre relaxation

Abstract

We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over a basic closed semialgebraic set, that is a polynomial optimization problem. These relaxations give an increasing sequence of lower bounds of the infimum. In this work we provide a new certificate for the optimal value of a Lasserre relaxation to be the optimal value of the polynomial optimization problem. This certificate is that a modified version of an optimal solution of the Lasserre relaxation is a generalized Hankel matrix. This certificate is more general than the already known certificate of an optimal solution being flat. In case that this modified matrix has a generalized Hankel form we will extract the potential minimizers with a truncated version of the Gelfand-Naimark-Segal construction on the optimal solution of the Lasserre relaxation. We prove also that the operators of this truncated construction commute if and only if the matrix of this modified optimal solution is a generalized Hankel matrix. This generalization of flatness will bring us to prove a result of Curto and Fialkow on the existence of quadrature rule if the optimal solution is flat and a result of Xu and Mysovskikh on the existence of a Gaussian quadrature rule if the modified optimal solution is a generalized Hankel matrix. We give a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem. Finally, we provide an experimental algorithm that in many cases gives us an upper bound of the global infimum of a real polynomial on R n . The algorithm that we present involves to solve a series of semidefinite programs whose feasible set is included in the feasible set of a moment relaxation. Our additional constraint try to provoke a flatness condition, like used by Curto and Fialkow, for the computed moments. At the end we present numerical results of the application of the algorithm to nonnegative polynomials which are not sums of squares. We also provide numerical results for the application of a version of this algorithm based on the method proposed by Nie, Demmel and Sturmfels.