Unrolled Neural Networks for Constrained Optimization

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. Unlike standard unrolling, which maps the updates of a prescribed iterative solver directly to network layers, CDU preserves the underlying optimization dynamics while learning the update mappings. The desired primal-dual dynamics are therefore induced through constrained learning by enforcing primal-descent and dual-ascent constraints. 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 establish theoretical guarantees on the feasibility and optimality of the coupled networks. We numerically evaluate the framework on quadratic programs (QPs) with linear constraints and power allocation in wireless networks.