Difference-of-convex Optimization Speeds Goemans-Williamson for Quadratic Unconstrained Binary Optimization Problems
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Solving complex optimisation problems frequently relies on the Goemans-Williamson procedure, a randomised technique for finding good solutions to challenging quadratic equations, but its computational demands can limit its application to large-scale problems. Now, Hadi Salloum, Roland Hildebrand, and Nhat Trung Nguyen, alongside colleagues from MIPT and Innopolis University, present a new method to significantly speed up this process. Their approach replaces the computationally intensive standard method with a difference-of-convex optimisation framework, efficiently approximating solutions and then refining them through direct expectation minimisation.
The team demonstrates that this technique not only achieves competitive results on real-world problems, such as those found in robotics and inverse kinematics, but also delivers substantial performance gains compared to existing state-of-the-art solvers.
Rank Constrained Relaxation for Optimization Problems Scientists have developed a new method to efficiently solve challenging optimization problems, including Max-Cut, Quadratic Programming, and Inverse Kinematics. The research introduces a rank-constrained relaxation technique combined with a specialized optimization solver, offering significant improvements in runtime compared to existing approaches. This method effectively reduces the complexity of the problem, allowing for faster computation without sacrificing solution quality. Experiments demonstrate that this approach consistently outperforms standard methods like Semidefinite Programming relaxations and common heuristics, particularly as problem size increases.
The team’s method iteratively solves a series of Second-Order Cone Programming (SOCP) problems to refine the solution. By carefully controlling the rank of a key matrix variable, the researchers reduce computational demands while maintaining accuracy. Different rank constraints, such as R5 and R10, generally provided the best balance between speed and accuracy. This innovative approach offers a substantial advancement in optimization, enabling solutions to larger and more complex problems than previously possible. Efficient QUBO Solving via DC Optimization Researchers have created a new method to accelerate the solution of quadratic unconstrained binary optimization (QUBO) problems, prevalent in fields like machine learning and robotics.
The team bypasses computationally expensive semi-definite programming by employing a difference-of-convex (DC) optimization framework. This allows for efficient approximation of solutions and generation of candidate vectors used within the Goemans-Williamson randomized rounding procedure to produce high-quality binary solutions. The method involves searching over a limited set of matrices with a rank of two or less, and directly minimizing the expected cost functional. Experiments on real-world QUBO instances, including inverse kinematics problems, demonstrate competitive approximation guarantees alongside substantial computational gains. This advancement offers a significant improvement in solving complex optimization challenges, paving the way for faster and more efficient solutions in diverse applications. DC Optimization Accelerates QUBO Problem Solving Researchers have developed a new method to accelerate the solution of quadratic unconstrained binary optimization (QUBO) problems, found in diverse applications like machine learning and robotics. The research focuses on improving the Goemans-Williamson (GW) randomized rounding procedure, a common approach to finding approximate solutions to these complex problems. Instead of relying on computationally expensive semi-definite programming, the team employed a difference-of-convex (DC) optimization framework to efficiently approximate solutions, significantly reducing processing time. The method involves generating candidate vectors using DC optimization and then applying these within the GW randomized rounding scheme to produce high-quality binary solutions. Experiments on both randomly generated and real-world QUBO instances demonstrate that the method achieves solution quality comparable to, or exceeding, that of established metaheuristic methods like Simulated Annealing and Tabu Search. This advancement offers a favorable trade-off between solution quality and computational efficiency for a range of applications.
Efficient Covariance Approximation for Quadratic Optimization Researchers have developed a new method for solving quadratic binary unconstrained optimization problems, building upon the established Goemans-Williamson randomized procedure.
The team developed a technique to efficiently approximate the covariance matrix required by this procedure, avoiding the computationally expensive step of solving a semi-definite relaxation. Instead of directly optimizing over the covariance matrix, the researchers optimized over its factor, located on a smooth manifold, allowing for greater flexibility in controlling the rank of the solution. The resulting covariance matrix demonstrates a better expected objective value compared to solutions derived from traditional semi-definite relaxation, while also being computationally cheaper to obtain. Experiments confirm that this improved covariance matrix leads to better solutions when used within the randomized rounding scheme. This approach offers a promising advancement in solving challenging optimization problems, potentially enabling more efficient solutions in fields such as machine learning and materials science. 👉 More information 🗞 Speeding up the Goemans-Williamson randomized procedure by difference-of-convex optimization 🧠 ArXiv: https://arxiv.org/abs/2512.08852 Tags: