Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity

TL;DR

This paper introduces Sparse-BSOS, a sparse hierarchy of semidefinite programs for large-scale polynomial optimization that exploits problem sparsity to ensure convergence to the global optimum.

Contribution

It develops a sparse version of the BSOS hierarchy that handles large problems with structured sparsity and guarantees convergence under certain conditions.

Findings

Converges to the global optimum for problems with the running intersection property.

Finite convergence for SOS-convex problems at the first hierarchy level.

Handles large-scale problems efficiently by fixing semidefinite constraint size.

Abstract

We provide a sparse version of the bounded degree SOS hierarchy BSOS [7] for polynomial optimization problems. It permits to treat large scale problems which satisfy a structured sparsity pattern. When the sparsity pattern satisfies the running intersection property this Sparse-BSOS hierarchy of semidefinite programs (with semidefinite constraints of fixed size) converges to the global optimum of the original problem. Moreover, for the class of SOS-convex problems, finite convergence takes place at the first step of the hierarchy, just as in the dense version.