On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

In this paper, we are interested in developing polynomial decomposition techniques to reformulate real valued multivariate polynomials into difference-of-sums-of-squares (namely, D-SOS) and difference-of-convex-sums-of-squares (namely, DC-SOS). Firstly, we prove that the set of D-SOS and DC-SOS polynomials are vector spaces and equivalent to the set of real valued polynomials. Moreover, the problem of finding D-SOS and DC-SOS decompositions are equivalent to semidefinite programs (SDP) which can be solved to any desired precision in polynomial time. Some important algebraic properties and the relationships among the set of sums-of-squares (SOS) polynomials, positive semidefinite (PSD) polynomials, convex-sums-of-squares (CSOS) polynomials, SOS-convex polynomials, D-SOS and DC-SOS polynomials are discussed. Secondly, we focus on establishing several practical algorithms for constructing D-SOS and DC-SOS decompositions for any polynomial without solving SDP. Using DC-SOS decomposition, we can reformulate polynomial optimization problems in the realm of difference-of-convex (DC) programming, which can be handled by efficient DC programming approaches. Some examples illustrate how to use our methods for constructing D-SOS and DC-SOS decompositions. Numerical performance of D-SOS and DC-SOS decomposition algorithms and their parallelized methods are tested on a synthetic dataset with 1750 randomly generated large and small sized sparse and dense polynomials. Some real-world applications in higher order moment portfolio optimization problems, eigenvalue complementarity problems, Euclidean distance matrix completion problems, and Boolean polynomial programs are also presented.Comment: 47 pages, 19 figure

Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis

This thesis settles a number of questions related to computational complexity and algebraic, semidefinite programming based relaxations in optimization and control.Comment: PhD Thesis, MIT, September, 201