Sparse Polynomial Matrix Optimization

TL;DR

This paper investigates how sparsity techniques can significantly reduce the computational complexity of polynomial matrix optimization problems, especially those involving sum-of-squares methods and semidefinite programming relaxations.

Contribution

It introduces novel sparsity exploitation methods for polynomial matrix inequalities, including term, correlative, and matrix sparsity, with theoretical convergence and optimality results.

Findings

Sparsity methods reduce semidefinite program sizes.

Convergence properties vary with sparsity type.

Effective in practical polynomial matrix optimization examples.

Abstract

A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of polynomial matrix inequalities. This work explores the use of sparsity methods in reducing the complexity of sum-of-squares based methods in verifying polynomial matrix inequalities or solving polynomial matrix optimization. In the unconstrained setting, Newton polytopes can be employed to sparsify the monomial basis, resulting in smaller semidefinite programs. In the general setting, we show how to exploit different types of sparsity (term sparsity, correlative sparsity, matrix sparsity) encoded in polynomial matrices to derive sparse semidefinite programming relaxations for polynomial matrix optimization. For term sparsity, we show that the block structures of the term sparsity iterations with maximal chordal extensions converge to the one determined by PMI sign symmetries. For correlative sparsity, unlike the scalar case, we provide a counterexample showing that asymptotic convergence does not hold under the Archimedean condition and the running intersection property. By employing the theory of matrix-valued measures, we establish several results on detecting global optimality and retrieving optimal solutions under correlative sparsity. The effectiveness of sparsity methods on reducing computational complexity is demonstrated on various examples of polynomial matrix optimization.