Projection, Degeneracy, and Singularity Degree for Spectrahedra

TL;DR

This paper explores facial reduction and semi-smooth Newton methods for projecting onto spectrahedra, revealing how degeneracy and ill-conditioning affect convergence, with applications to correlation matrices and combinatorial optimization relaxations.

Contribution

It establishes a connection between facial reduction, degeneracy, and the convergence behavior of Newton methods for spectrahedron projections, providing new formulas and empirical insights.

Findings

Ill-conditioning of the Jacobian indicates degeneracy in spectrahedron projections.

Derived an elegant formula for projection onto faces of the semidefinite cone.

Applied results to correlation matrices and semidefinite relaxations of combinatorial problems.

Abstract

Facial reduction, FR, is a regularization technique for convex programs where the strict feasibility constraint qualification, CQ, fails. Though this CQ holds generically, failure is pervasive in applications such as semidefinite relaxations of hard discrete optimization problems. In this paper we relate FR to the analysis of the convergence behaviour of a semi-smooth Newton root finding method for the projection onto a spectrahedron, i.e., onto the intersection of a linear manifold and the semidefinite cone. In the process, we derive and use an elegant formula for the projection onto a face of the semidefinite cone. We show further that the ill-conditioning of the Jacobian of the Newton method near optimality characterizes the degeneracy of the nearest point in the spectrahedron. We apply the results, both theoretically and empirically, to the problem of finding nearest points to the sets of: (i) correlation matrices or the elliptope; and (ii) semidefinite relaxations of permutation matrices or the vontope, i.e., the feasible sets for the semidefinite relaxations of the max-cut and quadratic assignment problems, respectively.