Conic optimization with spectral functions on Euclidean Jordan algebras

TL;DR

This paper develops new barrier functions for spectral cones in Euclidean Jordan algebras, enabling efficient conic optimization with spectral functions, and demonstrates their practical effectiveness in an open-source solver.

Contribution

It introduces simple, self-concordant barriers for spectral cones, along with stable evaluation procedures, and integrates them into a solver that outperforms existing methods on certain problems.

Findings

Barriers are self-concordant with nearly optimal parameters.

Hypatia solver outperforms MOSEK 9 on natural formulations.

Efficient, stable procedures for barrier evaluation are derived.

Abstract

Spectral functions on Euclidean Jordan algebras arise frequently in convex models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, i.e. proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones - the root-determinant cones and the matrix monotone derivative cones - we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equivalent extended formulations written using only the cones these solvers support.