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

arXiv (Cornell University)

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 $800 \times 800$.

Authors 3

1. Richard L. Zhu

   University of Chicago

   Affiliation as printed

   Department of Computational and Applied Mathematics University of Chicago Chicago , IL 60637

2. Mathias Oster Aachen Institute of Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute of Geometry and Practical Mathematics RWTH Aachen Aachen , Germany

3. Yuehaw Khoo

   University of Chicago

   Affiliation as printed

   Department of Statistics University of Chicago Chicago , IL 60637