Conic Optimization with Spectral Functions on Euclidean Jordan Algebras

Spectral functions on Euclidean Jordan algebras arise frequently in convex optimization models. Despite the success of primal-dual conic interior point solvers, there has been little work enabling direct support for spectral cones, that is, proper nonsymmetric cones defined from epigraphs and perspectives functions. We propose simple logarithmically homogeneous barriers we derive efficient, numerically stable procedures evaluating barrier oracles such as inverse Hessian operators. For two useful classes cones—the root-determinant matrix monotone derivative cones—we show are self-concordant, with nearly optimal parameters. implement these our open-source solver Hypatia, write simple, natural formulations four applied problems. Our computational benchmarks demonstrate Hypatia often solves more efficiently than advanced solvers MOSEK 9 solve equivalent extended written using only support. Funding: This was supported by Office Naval Research [Grant N00014-18-1-2079] National Science Foundation OAC-1835443].