Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization

TL;DR

This paper reviews recent advances in chordal and factor-width decompositions that enable scalable semidefinite and polynomial optimization, significantly improving computational efficiency for large-scale systems in control and machine learning.

Contribution

It provides a comprehensive overview of recent developments in decomposition methods, highlighting their connections, differences, and practical applications in large-scale optimization.

Findings

Chordal decomposition exploits matrix sparsity for efficiency.

Factor-width decomposition trades feasibility for tractability.

Significant computational savings demonstrated in control and machine learning problems.

Abstract

Chordal and factor-width decomposition methods for semidefinite programming and polynomial optimization have recently enabled the analysis and control of large-scale linear systems and medium-scale nonlinear systems. Chordal decomposition exploits the sparsity of semidefinite matrices in a semidefinite program (SDP), in order to formulate an equivalent SDP with smaller semidefinite constraints that can be solved more efficiently. Factor-width decompositions, instead, relax or strengthen SDPs with dense semidefinite matrices into more tractable problems, trading feasibility or optimality for lower computational complexity. This article reviews recent advances in large-scale semidefinite and polynomial optimization enabled by these two types of decomposition, highlighting connections and differences between them. We also demonstrate that chordal and factor-width decompositions allow for significant computational savings on a range of classical problems from control theory, and on more recent problems from machine learning. Finally, we outline possible directions for future research that have the potential to facilitate the efficient optimization-based study of increasingly complex large-scale dynamical systems.