S-SOS: Stochastic Sum-Of-Squares for Parametric Polynomial Optimization

Richard Licheng Zhu, Mathias Oster, Yuehaw Khoo

Keywords: convex optimization, semidefinite programming, sum-of-squares, uncertainty quantification, polynomial optimization

TL;DR: We discuss a stochastic sum-of-squares algorithm for global polynomial optimization and use it for uncertainty quantification on sensor network localization problems.

Abstract: Global polynomial optimization is an important tool across applied mathematics, with many applications in operations research, engineering, and the physical sciences. In various settings, the polynomials depend on external parameters that may be random. We discuss a stochastic sum-of-squares (S-SOS) algorithm based on the sum-of-squares hierarchy that constructs a series of semidefinite programs to jointly find strict lower bounds on the global minimum and extracts candidates for parameterized global minimizers. We prove quantitative convergence of the hierarchy as the degree increases and use it to solve unconstrained and constrained polynomial optimization problems parameterized by random variables. By employing n-body priors from condensed matter physics to induce sparsity, we can use S-SOS to produce solutions and uncertainty intervals for sensor network localization problems containing up to 40 variables and semidefinite matrix sizes surpassing 800 x 800.

Primary Area: Optimization (convex and non-convex, discrete, stochastic, robust)

Submission Number: 3394