Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction

A basic closed semialgebraic subset of $${\mathbb {R}}^{n}$$ R n is defined by simultaneous polynomial inequalities $$p_{1}\ge 0,\ldots ,p_{m}\ge 0$$ p 1 ≥ 0 , … , p m ≥ 0 . We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over such a set. These relaxations give an increasing sequence of lower bounds of the infimum. In this paper 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 to check if a certain matrix has a generalized Hankel form. This certificate is more general than the already known certificate of an optimal solution being flat. In case we have detected optimality 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 enable us to prove, with the use of the GNS truncated construction, 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 . At the end, we provide a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem.

Algorithmization and Software Implementation of the Method of Eliminating Variables in Polynomial Optimization Problems

The method of sequential exclusion of variables in polynomial optimization problems is considered. A number of problems are solved using this method. The practical steps of an algorithm are described, which reduces the initial polynomial optimization problem to a multi-stage branching process of obtaining a finite number of alternative problems, the output of which gives a finite set of polynomials in one variable. As a result, solving a number of polynomial problems reduces to sorting out a finite number of vectors whose components are the real roots of polynomials.

Sparse Sum-of-Squares Optimization for Model Updating Through Minimization of Modal Dynamic Residuals

This research investigates the application of sum-of-squares (SOS) optimization method on finite element model updating through minimization of modal dynamic residuals. The modal dynamic residual formulation usually leads to a nonconvex polynomial optimization problem, the global optimality of which cannot be guaranteed by most off-the-shelf optimization solvers. The SOS optimization method can recast a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem. However, the size of the SDP problem can grow very large, sometimes with hundreds of thousands of variables. To improve the computation efficiency, this study exploits the sparsity in SOS optimization to significantly reduce the size of the SDP problem. A numerical example is provided to validate the proposed method.

A Shifted Power Method for Homogenous Polynomial Optimization over Unit Spheres

In this paper, we propose a shifted power method for a type of polynomial optimization problem over unit spheres. The global convergence of the proposed method is established and an easily implemented scope of the shifted parameter is provided.

SDP RELAXATIONS FOR QUADRATIC OPTIMIZATION PROBLEMS DERIVED FROM POLYNOMIAL OPTIMIZATION PROBLEMS

Based on the convergent sequence of SDP relaxations for a multivariate polynomial optimization problem (POP) by Lasserre (2006), Waki et al. (2006) constructed a sequence of sparse SDP relaxations to solve sparse POPs efficiently. Nevertheless, the size of the sparse SDP relaxation is the major obstacle in order to solve POPs of higher degree. This paper proposes an approach to transform general POPs to quadratic optimization problems (QOPs), which allows to reduce the size of the SDP relaxation substantially. We introduce different heuristics resulting in equivalent QOPs and show how sparsity of a POP is maintained under the transformation procedure. As the most important issue, we discuss how to increase the quality of the SDP relaxation for a QOP. Moreover, we increase the accuracy of the solution of the SDP relaxation by applying additional local optimization techniques. Finally, we demonstrate the high potential of this approach through numerical results for large scale POPs of higher degree.