Error Bounds and Singularity Degree in Semidefinite Programming

Abstract

An important process in optimization is to determine the quality of a proposed solution. This usually entails calculation of the distance of a proposed solution to the optimal set and is referred to as forward error. Since the optimal set is not known, we generally view forward error as intractable. An alternative to forward error is to measure the violation in the constraints or optimality conditions. This is referred to as backward error and it is generally easy to compute. A major issue in optimization occurs when a proposed solution has small backward error, i.e., looks good to the user, but has large forward error, i.e., is far from the optimal set.

In 2001, Jos Sturm developed a remarkable upper bound on forward error for spectrahedra (optimal sets of semidefinite programs) in terms of backward error. His bound creates a hierarchy among spectrahedra that is based on singularity degree, an integer between 0 and n-1, derived from facial reduction. For problems with small singularity degree, forward error is similar to backward error, but this may not be true for problems with large singularity degree.

In this thesis we provide a method to obtain numerical lower bounds on forward error, thereby complimenting the bounds of Sturm. While the bounds of Sturm identify good convergence, our bounds allow us to detect poor convergence. Our approach may also be used to provide lower bounds on singularity degree, a measure that is difficult to compute in some instances. We show that large singularity degree leads to some undesirable convergence properties for a specific family of central paths.

We apply our results in a theoretical sense to some Toeplitz matrix completion problems and in a numerical sense to several test spectrahedra.