Controlling Continuous Relaxation for Combinatorial Optimization

Motivated by developments in machine learning technologies, unsupervised learning (UL)-based solvers for CO problems have recently been proposed. These solvers train a neural network that outputs a solution by optimizing the CO objective directly. UL-based solvers have several advantages over traditional methods. However, various studies have shown that these solvers underperform compared to greedy algorithms for complex CO problems. In addition, these solvers employ a continuous relaxation strategy; thus, post-learning rounding from the continuous space back to the original discrete space is required, undermining the robustness of the results. To address these problems, we propose the continuous relaxation annealing (CRA) strategy. The CRA introduces a penalty term to control the continuity and discreteness of the relaxed variables and eliminate local optima. In addition, the CRA implements an annealing process for the penalty term that initially prioritizes continuous solutions and progressively transitions towards discreet solutions until the relaxed variables become nearly discrete, eliminating the artificial rounding. Experimental results demonstrate that the CRA significantly enhances the UL-based solvers, outperforming both existing UL-based solvers and greedy algorithms for complex CO problems.