Projections onto Spectral Matrix Cones

TL;DR

This paper introduces a method to efficiently project matrices onto spectral matrix cones by reducing the problem to eigenvalue or singular value projections, significantly speeding up large-scale semidefinite programming.

Contribution

It demonstrates that spectral matrix cone projections can be efficiently computed via eigenvalue or singular value projections, enhancing first-order solvers like SCS.

Findings

Speedups of up to ten times in solving semidefinite programs.

Effective integration of spectral projections into SCS.

Applications in experimental design, PCA, and graph partitioning.

Abstract

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning.