Finding the best solution is the most common objective in combinatorial optimization (CO) problems. However, a single solution may not be suitable in practical scenarios, as the objective functions and constraints are only approximations of original real-world situations. To tackle this, finding (i) "heterogeneous solutions", diverse solutions with distinct characteristics, and (ii) "penalty-diversified solutions", variations in constraint severity, are natural directions. This strategy provides the flexibility to select a suitable solution during post-processing. However, discovering these diverse solutions is more challenging than identifying a single solution. To overcome this challenge, this study introduces Continual Tensor Relaxation Annealing (CTRA) for unsupervised-learning-based CO solvers. CTRA addresses various problems simultaneously by extending the continual relaxation approach, which transforms discrete decision variables into continual tensors. This method finds heterogeneous and penalty-diversified solutions through mutual interactions, where the choice of one solution affects the other choices. Numerical experiments show that CTRA enables UL-based solvers to find heterogeneous and penalty-diversified solutions much faster than existing UL-based solvers. Moreover, these experiments reveal that CTRA enhances the exploration ability.

翻译：寻找最优解是组合优化（CO）问题中最常见的目标。然而，在实际场景中单一解可能并不适用，因为目标函数和约束条件仅是原始现实情境的近似。为此，发现（i）"异构解"（具有不同特征的多样化解）和（ii）"惩罚分化解"（约束严厉程度的变化）是自然的研究方向。该策略提供了在后处理阶段选择合适解的灵活性。然而，发现这些多样化解比识别单一解更具挑战性。为克服这一挑战，本研究针对基于无监督学习的CO求解器提出了连续张量松弛退火（CTRA）。CTRA通过扩展连续松弛方法（将离散决策变量转化为连续张量）来同时处理多种问题。该方法通过互作用机制发现异构解与惩罚分化解——在此机制中，某个解的选择会影响其他解的决策。数值实验表明，CTRA能使基于UL的求解器以远超现有UL求解器的速度找到异构解与惩罚分化解。此外，这些实验还揭示了CTRA对探索能力的增强效应。