Speeding up the Goemans-Williamson randomized procedure by difference-of-convex optimization

TL;DR

This paper introduces a difference-of-convex optimization approach to accelerate the Goemans-Williamson randomized procedure for solving quadratic binary problems, achieving high-quality solutions more efficiently.

Contribution

It proposes a novel DC optimization framework to approximate SDP solutions, replacing expensive semi-definite programming in the GW method, and demonstrates improved computational efficiency.

Findings

Achieves comparable approximation guarantees to state-of-the-art methods.

Significantly reduces computational time on real-world QUBO instances.

Outperforms quantum-inspired algorithms in benchmark tests.

Abstract

We present a novel approach to accelerate the Goemans-Williamson (GW) randomized rounding procedure for quadratic unconstrained binary optimization (QUBO) problems. Instead of solving the conventional semi-definite programming (SDP) relaxation, which is computationally expensive, we employ a difference-of-convex (DC) optimization framework to efficiently approximate the SDP solution. The DC optimization produces candidate vectors that are then used within the GW randomized rounding scheme to generate high-quality binary solutions. Furthermore, we perform direct expectation minimization over manifolds of matrices with limited rank to further enhance the solution quality. Our method is benchmarked on real-world QUBO instances, including inverse kinematics problems, and compared against state-of-the-art solvers, such as quantum-inspired algorithms, demonstrating competitive approximation guarantees alongside substantial computational gains.