Spectral Outer-Approximation Algorithms for Binary Semidefinite Problems

TL;DR

This paper introduces a spectral outer-approximation algorithm specifically designed for binary semidefinite programs derived from binary quadratically constrained quadratic programs, demonstrating competitive performance with existing solvers.

Contribution

It develops a novel spectral outer-approximation method that exploits the structure of BSDPs from BQCQPs, filling a gap in specialized ISDP algorithms.

Findings

The proposed algorithm is competitive with state-of-the-art ISDP solvers.

It outperforms some existing solvers in certain cases.

The approach highlights the potential of ISDP for solving BQCQPs.

Abstract

Integer semidefinite programming (ISDP) has recently gained attention due to its connection to binary quadratically constrained quadratic programs (BQCQPs), which can be exactly reformulated as binary semidefinite programs (BSDPs). However, it remains unclear whether this reformulation effectively uses existing ISDP solvers to address BQCQPs. To the best of our knowledge, no specialized ISDP algorithms exploit the unique structure of BSDPs derived from BQCQPs. This paper proposes a novel spectral outer approximation algorithm tailored for BSDPs derived from BQCQP reformulations. Our approach is inspired by polyhedral and second-order representable regions that outer approximate the feasible set of a semidefinite program relying on a spectral decomposition of a matrix that simultaneously diagonalizes the objective matrix and an aggregation of the constraint matrices. Computational experiments show that our algorithm is competitive with, and in some cases outperforms, state-of-the-art ISDP solvers such as SCIP-SDP and PAJARITO, highlighting ISDP's potential for solving BQCQPs.