Recent works in learning-integrated optimization have shown promise in settings where the optimization problem is only partially observed or where general-purpose optimizers perform poorly without expert tuning. By learning an optimizer $\mathbf{g}$ to tackle these challenging problems with $f$ as the objective, the optimization process can be substantially accelerated by leveraging past experience. The optimizer can be trained with supervision from known optimal solutions or implicitly by optimizing the compound function $f\circ \mathbf{g}$. The implicit approach may not require optimal solutions as labels and is capable of handling problem uncertainty; however, it is slow to train and deploy due to frequent calls to optimizer $\mathbf{g}$ during both training and testing. The training is further challenged by sparse gradients of $\mathbf{g}$, especially for combinatorial solvers. To address these challenges, we propose using a smooth and learnable Landscape Surrogate $M$ as a replacement for $f\circ \mathbf{g}$. This surrogate, learnable by neural networks, can be computed faster than the solver $\mathbf{g}$, provides dense and smooth gradients during training, can generalize to unseen optimization problems, and is efficiently learned via alternating optimization. We test our approach on both synthetic problems, including shortest path and multidimensional knapsack, and real-world problems such as portfolio optimization, achieving comparable or superior objective values compared to state-of-the-art baselines while reducing the number of calls to $\mathbf{g}$. Notably, our approach outperforms existing methods for computationally expensive high-dimensional problems.

翻译：近期关于学习与优化相结合的研究在以下场景中展现出潜力：优化问题仅被部分观测，或通用优化器未经专家调参时性能不佳。通过学习一个以$f$为目标函数的优化器$\mathbf{g}$来处理这些具有挑战性的问题，可以利用过往经验显著加速优化过程。该优化器可通过已知最优解的监督训练，或隐式地通过优化复合函数$f\circ \mathbf{g}$进行学习。隐式方法无需以最优解作为标签，且能处理问题的不确定性；然而，由于在训练和测试阶段需频繁调用优化器$\mathbf{g}$，其训练与部署过程较为缓慢。此外，$\mathbf{g}$的梯度稀疏性（尤其对于组合求解器）进一步增加了训练难度。为应对这些挑战，我们提出使用光滑且可学习的景观代理$M$替代$f\circ \mathbf{g}$。该代理可通过神经网络学习，其计算速度优于求解器$\mathbf{g}$，在训练中提供稠密光滑的梯度，能泛化至未见过的优化问题，并通过交替优化实现高效学习。我们在合成问题（包括最短路径问题和多维背包问题）以及实际应用（如投资组合优化）中测试了该方法，在减少$\mathbf{g}$调用次数的同时，取得了与最先进基线方法相当或更优的目标函数值。值得注意的是，本方法在处理计算代价高昂的高维问题时优于现有方法。