Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

TL;DR

Sieve-SDP is a straightforward facial reduction method for semidefinite programs that simplifies problems, detects infeasibility, and reduces size without relying on solvers, demonstrated through extensive computational testing.

Contribution

Introduces Sieve-SDP, a simple, solver-independent facial reduction algorithm for preprocessing SDPs, enhancing efficiency and infeasibility detection.

Findings

Effectively reduces SDP problem size

Detects infeasibility in many cases

Requires only Cholesky factorization

Abstract

We introduce Sieve-SDP, a simple facial reduction algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP inspects the constraints of the problem to detect lack of strict feasibility, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented in a few lines of code in machine precision. We present extensive computational results on several problem collections from the literature, with many SDPs coming from polynomial optimization.