Sums of Squares and Sparse Semidefinite Programming

Semidefinite Programming and Sums of Hermitian Squares of Noncommutative Polynomials

An algorithm for finding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the “Newton chip method”, a noncommutative analog of the classical Newton polytope method, and semidefinite programming.

Title: Moments, Sums of Squares and Semidefinite Programming

We introduce the generalized problem of moments (GPM) which has developments and impact in various area of Mathematics like algebra, Fourier analysis, functional analysis, operator theory, probability and statistics, to cite a few. In addition, and despite its rather simple and short formulation, the GPM has a large number of important applications in various fields like optimization, probabili...

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

Let G be a nite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T . Our combinatorial condition involves constructing a chordal c...

Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity

Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are ob...

Symmetry Groups, Semidefinite Programs, and Sums of Squares