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

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.

The background method: Theory and computations

“…This is convenient because currently available SDP solvers can handle multiple smaller LMIs more efficiently than a single large one. Similarly, the structure of each block can be exploited using chordal decomposition techniques for SDPs [89][90][91][92][93], which decompose each block into smaller ones by considering dense principal submatrices (see the bottom panel in Figure 1 for an illustration). Chordal decomposition, for this particular example and in general, can significantly reduce computational cost if these dense principal submatrices are small, enabling one to accurately resolve very sharp and possibly nonsmooth boundary layers [42,50].…”