A simple preprocessing algorithm for semidefinite programming

TL;DR

This paper introduces a straightforward preprocessing algorithm for semidefinite programming that simplifies problem constraints, reduces matrix size, and can detect infeasibility without relying on complex solvers, using only Cholesky factorization.

Contribution

The authors present a simple, solver-independent preprocessing method for semidefinite programs that efficiently reduces problem size and detects infeasibility.

Findings

Effective in reducing problem size

Detects infeasibility in many cases

Requires only Cholesky factorization

Abstract

We propose a very simple preprocessing algorithm for semidefinite programming. Our algorithm inspects the constraints of the problem, deletes redundant rows and columns in the constraints, and reduces the size of the variable matrix. It often detects infeasibility. Our algorithm does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented with a few lines of code in machine precision. We present computational results on a set of problems arising mostly from polynomial optimization.