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

TL;DR

The paper introduces S-SOS, a stochastic sum-of-squares hierarchy that efficiently solves parametric polynomial optimization problems with uncertainty, providing convergence guarantees and practical solutions for large-scale applications.

Contribution

It develops a novel stochastic sum-of-squares method with convergence analysis, enabling scalable solutions to parametric polynomial optimization problems with uncertainty.

Findings

Successfully applied to sensor network localization with up to 40 variables.

Produced solutions and uncertainty intervals for large semidefinite programs.

Demonstrated convergence of the hierarchy as polynomial degree increases.

Abstract

Global polynomial optimization is an important tool across applied mathematics, with many applications in operations research, engineering, and 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 extract 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 $800 \times 800$.