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.
@article{arxiv.2512.08852,
title = {Speeding up the Goemans-Williamson randomized procedure by difference-of-convex optimization},
author = {Hadi Salloum and Roland Hildebrand and Nhat Trung Nguyen and Vitali Pirau and Amer Al Badr and Mohammad Alkousa and Alexander Gasnikov},
journal= {arXiv preprint arXiv:2512.08852},
year = {2025}
}