Abstract--Embedding deep neural networks (NNs) into mixed-integer programs (MIPs) is attractive for decision making with learned constraints, yet state-of-the-art "monolithic" linearisa-tions blow up in size and quickly become intractable. In this paper, we introduce a novel dual-decomposition framework that relaxes the single coupling equality u = x with an augmented Lagrange multiplier and splits the problem into a vanilla MIP and a constrained NN block. Each part is tackled by the solver that suits it best--branch & cut for the MIP subproblem, first-order optimisation for the NN subproblem--so the model remains modular, the number of integer variables never grows with network depth, and the per-iteration cost scales only linearly with the NN size. LIB benchmark, our method proves scalable, modular, and adaptable: it runs 120 faster than an exact Big-M formulation on the largest test case; the NN sub-solver can be swapped from a log-barrier interior step to a projected-gradient routine with no code changes and identical objective value; and swapping the MLP for an LSTM backbone still completes the full optimisation in 47s without any bespoke adaptation. Intelligent decision systems increasingly integrate neural networks into decision-making and optimization pipelines [1-3].

We thank all reviewers for the valuable advice and questions. Our responses are provided below. Thank you for your valuable suggestions. We are sorry for the typo causing confusions in Assumption 2.2. Then, the dual multiplier updating step in Eq. (5) can be viewed as a one-step dual ascent in an online We will add this discussion in our paper.

We consider resource allocation problems in multi-user wireless networks, where the goal is to optimize a network-wide utility function subject to constraints on the ergodic average performance of users. We demonstrate how a state-augmented graph neural network (GNN) parametrization for the resource allocation policy circumvents the drawbacks of the ubiquitous dual subgradient methods by representing the network configurations (or states) as graphs and viewing dual variables as dynamic inputs to the model, viewed as graph signals supported over the graphs. Lagrangian maximizing state-augmented policies are learned during the offline training phase, and the dual variables evolve through gradient updates while executing the learned state-augmented policies during the inference phase. Our main contributions are to illustrate how near-optimal initialization of dual multipliers for faster inference can be accomplished with dual variable regression, leveraging a secondary GNN parametrization, and how maximization of the Lagrangian over the multipliers sampled from the dual descent dynamics substantially improves the training of state-augmented models. We demonstrate the superior performance of the proposed algorithm with extensive numerical experiments in a case study of transmit power control. Finally, we prove a convergence result and an exponential probability bound on the excursions of the dual function (iterate) optimality gaps.

Network slicing is a key feature in 5G/NG cellular networks that creates customized slices for different service types with various quality-of-service (QoS) requirements, which can achieve service differentiation and guarantee service-level agreement (SLA) for each service type. In Wi-Fi networks, there is limited prior work on slicing, and a potential solution is based on a multi-tenant architecture on a single access point (AP) that dedicates different channels to different slices. In this paper, we define a flexible, constrained learning framework to enable slicing in Wi-Fi networks subject to QoS requirements. We specifically propose an unsupervised learning-based network slicing method that leverages a state-augmented primal-dual algorithm, where a neural network policy is trained offline to optimize a Lagrangian function and the dual variable dynamics are updated online in the execution phase. We show that state augmentation is crucial for generating slicing decisions that meet the ergodic QoS requirements.