Optimization problems with nonlinear cost functions and combinatorial constraints appear in many real-world applications but remain challenging to solve efficiently compared to their linear counterparts. To bridge this gap, we propose $\textbf{SurCo}$ that learns linear $\underline{\text{Sur}}$rogate costs which can be used in existing $\underline{\text{Co}}$mbinatorial solvers to output good solutions to the original nonlinear combinatorial optimization problem. The surrogate costs are learned end-to-end with nonlinear loss by differentiating through the linear surrogate solver, combining the flexibility of gradient-based methods with the structure of linear combinatorial optimization. We propose three $\texttt{SurCo}$ variants: $\texttt{SurCo}-\texttt{zero}$ for individual nonlinear problems, $\texttt{SurCo}-\texttt{prior}$ for problem distributions, and $\texttt{SurCo}-\texttt{hybrid}$ to combine both distribution and problem-specific information. We give theoretical intuition motivating $\texttt{SurCo}$, and evaluate it empirically. Experiments show that $\texttt{SurCo}$ finds better solutions faster than state-of-the-art and domain expert approaches in real-world optimization problems such as embedding table sharding, inverse photonic design, and nonlinear route planning.

翻译：具有非线性成本函数和组合约束的优化问题广泛应用于现实场景，但相较于线性问题，其高效求解仍具挑战性。为弥合这一差距，我们提出$\textbf{SurCo}$方法，该方法通过学习线性$\underline{\text{代理}}$成本，可将其嵌入现有$\underline{\text{组合}}$求解器中，为原始非线性组合优化问题生成优质解。该代理成本通过线性代理求解器的可微化处理，结合非线性损失函数进行端到端学习，融合了梯度方法的灵活性与线性组合优化的结构特性。我们提出三种$\texttt{SurCo}$变体：针对单一非线性问题的$\texttt{SurCo}-\texttt{zero}$、面向问题分布的$\texttt{SurCo}-\texttt{prior}$，以及融合分布与问题特定信息的$\texttt{SurCo}-\texttt{hybrid}$。本文提供$\texttt{SurCo}$的理论直觉支撑，并进行实证评估。实验表明，在嵌入表分片、逆向光子设计及非线性路径规划等实际优化问题中，$\texttt{SurCo}$相比现有最优方法和领域专家方法，能以更快速度找到更优解。