Finding the best solution is a common objective in combinatorial optimization (CO). In practice, directly handling constraints is often challenging, incorporating them into the objective function as the penalties. However, balancing these penalties to achieve the desired solution is time-consuming. Additionally, formulated objective functions and constraints often only approximate real-world scenarios, where the optimal solution is not necessarily the best solution for the original real-world problem. One solution is to obtain (i) penalty-diversified solutions with varying penalty strengths for the former issue and (ii) variation-diversified solutions with different characteristics for the latter issue. Users can then post-select the desired solution from these diverse solutions. However, efficiently finding these diverse solutions is more difficult than identifying one. This study introduces Continual Tensor Relaxation Annealing (CTRA) for unsupervised-learning (UL)-based CO solvers, a computationally efficient framework for finding these diverse solutions in a single training run. The key idea is to leverage representation learning capability to automatically and efficiently learn common representations and parallelization. Numerical experiments show that CTRA enables UL-based solvers to find these diverse solutions much faster than repeatedly running existing UL-based solvers.

翻译：在组合优化（CO）中，寻找最优解是一个常见目标。实践中，直接处理约束条件通常具有挑战性，因此常将其作为惩罚项纳入目标函数。然而，平衡这些惩罚项以获得期望解的过程十分耗时。此外，形式化的目标函数与约束条件往往只是对现实场景的近似，此时最优解未必是原始现实问题的最佳解。针对前者，可通过获取（i）具有不同惩罚强度的惩罚多样化解；针对后者，可通过获取（ii）具有不同特征的变异多样化解来解决。用户随后可从这些多样化解中筛选出所需解。然而，高效寻找这些多样化解比单一解的识别更为困难。本研究针对基于无监督学习（UL）的组合优化求解器，提出连续张量松弛退火（CTRA）框架，通过单次训练即可高效计算获得多样化解。其核心思想是利用表征学习能力，自动高效地学习通用表征并实现并行化。数值实验表明，CTRA能使基于无监督学习的求解器，比重复运行现有同类方法更快地找到这些多样化解。