# Error Bounds and Singularity Degree in Semidefinite Programming

## TL;DR

This paper explores the relationship between error bounds, singularity degree, and convergence speed in semidefinite programming, providing new bounds and insights supported by numerical evidence.

## Contribution

It introduces a method to bound the maximum rank of solutions and establishes a link between singularity degree and convergence rate, complementing Sturm's upper bound.

## Key findings

- Large singularity degree can cause slow convergence.
- A lower bound on forward error is derived for certain sequences.
- Numerical results support the theoretical findings.

## Abstract

In semidefinite programming a proposed optimal solution may be quite poor in spite of having sufficiently small residual in the optimality conditions. This issue may be framed in terms of the discrepancy between forward error (the unmeasurable `true error') and backward error (the measurable violation of optimality conditions). In his seminal work, Sturm provided an upper bound on forward error in terms of backward error and singularity degree. In this paper we provide a method to bound the maximum rank over all solutions and use this result to obtain a lower bound on forward error for a class of convergent sequences. This lower bound complements the upper bound of Sturm. The results of Sturm imply that semidefinite programs with slow convergence necessarily have large singularity degree. Here we show that large singularity degree is, in some sense, also a sufficient condition for slow convergence for a family of external-type `central' paths. Our results are supported by numerical observations.