A Two-Step Pre-Processing for Semidefinite Programming

TL;DR

This paper proposes a two-step pre-processing approach for semidefinite programming that combines chordal-completion and facial reduction, leading to improved computational efficiency on benchmark and structured instances.

Contribution

It demonstrates that applying facial reduction after chordal-completion enhances SDP solving performance compared to traditional methods.

Findings

Two-step pre-processing outperforms standard interior-point methods.

Combining chordal-completion with facial reduction reduces problem size effectively.

Experimental results on benchmark and structured instances confirm efficiency gains.

Abstract

In semidefinite programming (SDP), a number of pre-processing techniques have been developed including chordal-completion procedures, which reduce the dimension of individual constraints by exploiting sparsity therein, and facial reduction, which reduces the dimension of the problem by removing redundant rows and columns. This paper suggest that these work in a complementary manner and that facial reduction should be used after chordal-completion procedures. In computational experiments on SDP instances from the SDPLib, a benchmark, and structured instances from polynomial and binary quadratic optimisation, we show that such two-step pre-processing with a standard interior-point method outperforms the interior point method, with or without the traditional pre-processing.