[论文解读] Unrolled Neural Networks for Constrained Optimization

Samar Hadou, Alejandro R. Ribeiro|arXiv (Cornell University)|Jan 24, 2026

Stochastic Gradient Optimization Techniques被引用 0

一句话总结

本论文提出受约束的对偶展开（CDU），一对可学习的原问题与对偶网络，通过在对偶域展开受约束优化，以实现近最优、近可行解并具有强鲁棒性外推（OOD）泛化能力。训练采用交替方案，结合下降/上升约束以模拟对偶上升动力学。

ABSTRACT

In this paper, we develop unrolled neural networks to solve constrained optimization problems, offering accelerated, learnable counterparts to dual ascent (DA) algorithms. Our framework, termed constrained dual unrolling (CDU), comprises two coupled neural networks that jointly approximate the saddle point of the Lagrangian. The primal network emulates an iterative optimizer that finds a stationary point of the Lagrangian for a given dual multiplier, sampled from an unknown distribution. The dual network generates trajectories towards the optimal multipliers across its layers while querying the primal network at each layer. Departing from standard unrolling, we induce DA dynamics by imposing primal-descent and dual-ascent constraints through constrained learning. We formulate training the two networks as a nested optimization problem and propose an alternating procedure that updates the primal and dual networks in turn, mitigating uncertainty in the multiplier distribution required for primal network training. We numerically evaluate the framework on mixed-integer quadratic programs (MIQPs) and power allocation in wireless networks. In both cases, our approach yields near-optimal near-feasible solutions and exhibits strong out-of-distribution (OOD) generalization.

研究动机与目标

• 将基于模型的优化与数据驱动学习结合用于受约束问题。
• 开发CDU，两组耦合的展开网络，模拟对偶上升动力学。
• 在训练中施加下降与上升约束，以提高稳定性和泛化能力。
• 在混合整数二次规划（MIQP）和无线功率分配任务上验证有效性。
• 提供一种训练框架，能够通过交替优化处理未知的乘子分布。

提出的方法

• 使用拉格朗日函数及对偶函数（L(x, lambda; z), D*: max over lambda）将受约束优化问题对偶化。
• 设计两组展开网络：原问题Phi_P在给定λ时近似拉格朗日函数的极小值点，对偶Phi_D通过每层查询原网络来更新λ。
• 在训练时对原层实现拉格朗日函数的单调下降，对偶层通过约束松弛实现对偶函数的单调上升（通过约束松弛）。
• 采用嵌套的交替训练方案：外层训练对偶，内层训练原问题，迭代采样乘子分布。
• 采用无监督、数据驱动的方法，通过最小化并约束对一组问题实例分布（z）和乘子分布的期望拉格朗日损失来训练。
• 使用图神经网络建模网络，利用结构性和可迁移性。

Figure 1 : Trajectories generated by (left) DA algorithm, (middle) constrained dual unrolling and (right) its unconstrained counterpart for a QP instance: (Top) primal trajectories toward the stationary point of the Lagrangian ${\mathcal{L}}(\cdot,\bm{\lambda};{\mathbf{z}})$ , and (bottom) dual traj

实验结果

研究问题

• RQ1 CDU是否能够将原始网络引导到模拟约束问题的对偶上升动力学？
• RQ2 施加下降/上升约束是否能提升稳定性和对OOD的泛化能力？
• RQ3 展开的原/对偶网络在不同问题实例上多大程度上能够逼近驻点和最优乘子？
• RQ4 该方法在MIQP和无线功率分配任务上是否有效，且无需事先给定精确的乘子分布？

主要发现

• CDU在MIQP和无线功率分配任务中产生近最优、近可行的解。
• 施加下降与上升约束提高了终层的准确性并改善了对OOD的泛化。
• 对偶网络的轨迹被引导以最大化对偶函数，而原网络收敛到拉格朗日函数的驻点。
• 一种交替训练方案有效采样当前对偶网络生成的乘子分布。
• 该框架对具体原/对偶架构及问题类别具有无关性，并展示了图神经网络的使用。
• 结果显示实现了对传统对偶上升方法的加速、可学习化的替代。

Figure 3 : Performance of constrained dual unrolling across $14$ layers vs an iterative DA algorithm. (Left) The distance to the primal optimum ${\mathbf{x}}^{*}$ , (middle) the distance to the dual optimum $\bm{\lambda}^{*}$ , and (right) the objective function (a measure of optimality). The 14-lay