Decision-focused learning (DFL) offers an end-to-end approach to the predict-then-optimize (PO) framework by training predictive models directly on decision loss (DL), enhancing decision-making performance within PO contexts. However, the implementation of DFL poses distinct challenges. Primarily, DL can result in deviation from the physical significance of the predictions under limited data. Additionally, some predictive models are non-differentiable or black-box, which cannot be adjusted using gradient-based methods. To tackle the above challenges, we propose a novel framework, Decision-Focused Fine-tuning (DFF), which embeds the DFL module into the PO pipeline via a novel bias correction module. DFF is formulated as a constrained optimization problem that maintains the proximity of the DL-enhanced model to the original predictive model within a defined trust region. We theoretically prove that DFF strictly confines prediction bias within a predetermined upper bound, even with limited datasets, thereby substantially reducing prediction shifts caused by DL under limited data. Furthermore, the bias correction module can be integrated into diverse predictive models, enhancing adaptability to a broad range of PO tasks. Extensive evaluations on synthetic and real-world datasets, including network flow, portfolio optimization, and resource allocation problems with different predictive models, demonstrate that DFF not only improves decision performance but also adheres to fine-tuning constraints, showcasing robust adaptability across various scenarios.

Neural networks (NNs) have emerged as promising tools for solving constrained optimization problems in real-time. However, ensuring constraint satisfaction for NN-generated solutions remains challenging due to prediction errors. Existing methods to ensure NN feasibility either suffer from high computational complexity or are limited to specific constraint types. We present Bisection Projection, an efficient approach to ensure NN solution feasibility for optimization over general compact sets with non-empty interiors. Our method comprises two key components: (i) a dedicated NN (called IPNN) that predicts interior points (IPs) with low eccentricity, which naturally accounts for approximation errors; (ii) a bisection algorithm that leverages these IPs to recover solution feasibility when initial NN solutions violate constraints. We establish theoretical guarantees by providing sufficient conditions for IPNN feasibility and proving bounded optimality loss of the bisection operation under IP predictions. Extensive evaluations on real-world non-convex problems demonstrate that Bisection Projection achieves superior feasibility and computational efficiency compared to existing methods, while maintaining comparable optimality gaps.

Projection-free first-order methods, e.g., the celebrated Frank-Wolfe (FW) algorithms, have emerged as powerful tools for optimization over simple convex sets such as polyhedra, because of their scalability, fast convergence, and iteration-wise feasibility without costly projections. However, extending these methods effectively to general compact convex sets remains challenging and largely open, as FW methods rely on expensive linear optimization oracles (LOO), while penalty-based methods often struggle with poor feasibility. We tackle this open challenge by presenting **Hom-PGD**, a novel projection-free method without expensive (optimization) oracles. Our method constructs a homeomorphism between the convex constraint set and a unit ball, transforming the original problem into an equivalent ball-constrained formulation, thus enabling efficient gradient-based optimization while preserving the original problem structure. We prove that Hom-PGD attains *optimal* convergence rates matching gradient descent with constant step-size to find an $\epsilon$-approximate (stationary) solution: $\mathcal{O}(\log (1/\epsilon))$ for strongly convex objectives, $\mathcal{O}(\epsilon^{-1})$ for convex objectives, and $\mathcal{O}(\epsilon^{-2})$ for non-convex objectives. Meanwhile, Hom-PGD enjoys a low per-iteration complexity of $\mathcal{O}(n^2)$, without expensive oracles like LOO or projection, where $n$ is the input size. Our framework further extends to certain non-convex sets, broadening its applicability in practical optimization scenarios with complex constraints. Extensive numerical experiments demonstrate that Hom-PGD achieves comparable convergence rates to state-of-the-art projection-free methods, while significantly reducing per-iteration runtime (up to 5 orders of magnitude faster) and thus the total problem-solving time.

By employing neural networks (NN) to learn input-solution mappings and passing a new input through the learned mapping to obtain a solution instantly, recent studies have shown remarkable speed improvements over iterative algorithms for solving optimization problems. Meanwhile, they also highlight methodological challenges to be addressed. In particular, general non-convex problems often present multiple optimal solutions for identical inputs, signifying a complex, multi-valued input-solution mapping. Conventional learning techniques, primarily tailored to learn single-valued mappings, struggle to train NNs to accurately decipher multi-valued ones, leading to inferior solutions. We address this fundamental issue by developing a generative learning approach using a rectified flow (RectFlow) model built upon ordinary differential equations. In contrast to learning input-solution mapping, we learn the mapping from input to solution distribution, exploiting the universal approximation capability of the RectFlow model. Upon receiving a new input, we employ the trained RectFlow model to sample high-quality solutions from the input-dependent distribution it has learned. Our approach outperforms conceivable GAN and Diffusion models in terms of training stability and run-time complexity. We provide a detailed characterization of the optimality loss and runtime complexity associated with our generative approach. Simulation results for solving non-convex problems show that our method achieves significantly better solution optimality than recent NN schemes, with comparable feasibility and speedup performance.

Since its debut in 2016, ResNet has become arguably the most favorable architecture in deep neural network (DNN) design. It effectively addresses the gradient vanishing/exploding issue in DNN training, allowing engineers to fully unleash DNN's potential in tackling challenging problems in various domains. Despite its practical success, an essential theoretical question remains largely open: how well/best can ResNet approximate functions? In this paper, we answer this question for several important function classes, including polynomials and smooth functions. In particular, we show that ResNet with constant width can approximate Lipschitz continuous function with a Lipschitz constant $\mu$ using $\mathcal{O}(c(d)(\varepsilon/\mu)^{-d/2})$ tunable weights, where $c(d)$ is a constant depending on the input dimension $d$ and $\epsilon>0$ is the target approximation error. Further, we extend such a result to Lebesgue-integrable functions with the upper bound characterized by the modulus of continuity. These results indicate a factor of $d$ reduction in the number of tunable weights compared with the classical results for ReLU networks. Our results are also order-optimal in $\varepsilon$, thus achieving optimal approximation rate, as they match a generalized lower bound derived in this paper. This work adds to the theoretical justifications for ResNet's stellar practical performance.

There has been growing interest in employing neural networks (NNs) to directly solve constrained optimization problems with low run-time complexity. However, it is non-trivial to ensure NN solutions strictly satisfy problem constraints due to inherent NN prediction errors. Existing feasibility-ensuring methods are either computationally expensive or lack performance guarantee. In this paper, we propose Homeomorphic Projection as a low-complexity scheme to guarantee NN solution feasibility for optimization over a general set homeomorphic to a unit ball, covering all compact convex sets and certain classes of non-convex sets. The idea is to (i) learn a minimum distortion homeomorphic mapping between the constraint set and a unit ball using a bi-Lipschitz invertible NN (INN), and then (ii) perform a simple bisection operation concerning the unit ball such that the INN-mapped final solution is feasible with respect to the constraint set with minor distortion-induced optimality loss. We prove the feasibility guarantee and bounded optimality loss under mild conditions. Simulation results, including those for non-convex AC-OPF problems in power grid operation, show that homeomorphic projection outperforms existing methods in solution feasibility and run-time complexity while achieving similar optimality loss.

There has been growing interest in employing neural network (NN) to directly solve constrained optimization problems with low run-time complexity. However, it is non-trivial to ensure NN solutions strictly satisfying problem constraints due to inherent NN prediction errors. Existing feasibility-ensuring methods either are computationally expensive or lack performance guarantee. In this paper, we propose homeomorphic projection as a low-complexity scheme to guarantee NN solution feasibility for optimization over a general set homeomorphic to a unit ball, covering all compact convex sets and certain classes of nonconvex sets. The idea is to (i) learn a minimum distortion homeomorphic mapping between the constraint set and a unit ball using an invertible NN (INN), and then (ii) perform a simple bisection operation concerning the unit ball so that the INN-mapped final solution is feasible with respect to the constraint set with minor distortion-induced optimality loss. We prove the feasibility guarantee and bound the optimality loss under mild conditions. Simulation results, including those for non-convex AC-OPF problems in power grid operation, show that homeomorphic projection outperforms existing methods in solution feasibility and run-time complexity, while achieving similar optimality loss.

Efficient traffic signal control is an important means to alleviate urban traffic congestion. Reinforcement learning (RL) has shown great potentials in devising optimal signal plans that can adapt to dynamic traffic congestion. However, several challenges still need to be overcome. Firstly, a paradigm of state, action, and reward design is needed, especially for an optimality-guaranteed reward function. Secondly, the generalization of the RL algorithms is hindered by the varied topologies and physical properties of intersections. Lastly, enhancing the cooperation between intersections is needed for large network applications. To address these issues, the Option-Action RL framework for universal Multi-intersection control (OAM) is proposed. Based on the well-known cell transmission model, we first define a lane-cell-level state to better model the traffic flow propagation. Based on this physical queuing dynamics, we propose a regularized delay as the reward to facilitate temporal credit assignment while maintaining the equivalence with minimizing the average travel time. We then recapitulate the phase actions as the constrained combinations of lane options and design a universal neural network structure to realize model generalization to any intersection with any phase definition. The multiple-intersection cooperation is then rigorously discussed using the potential game theory. We test the OAM algorithm under four networks with different settings, including a city-level scenario with 2,048 intersections using synthetic and real-world datasets. The results show that the OAM can outperform the state-of-the-art controllers in reducing the average travel time.

Balancing the supply and demand for ride-sourcing companies is a challenging issue, especially with real-time requests and stochastic traffic conditions of large-scale congested road networks. To tackle this challenge, this article proposes a robust and scalable approach that integrates reinforcement learning (RL) and a centralized programming (CP) structure to promote real-time taxi operations. Both real-time order matching decisions and vehicle relocation decisions at the microscopic network scale are integrated within a Markov decision process framework. The RL component learns the decomposed state-value function, which represents the taxi drivers' experience, the off-line historical demand pattern, and the traffic network congestion. The CP component plans nonmyopic decisions for drivers collectively under the prescribed system constraints to explicitly realize cooperation. Furthermore, to circumvent sparse reward and sample imbalance problems over the microscopic road network, this article proposed a temporal-difference learning algorithm with prioritized gradient descent and adaptive exploration techniques. A simulator is built and trained with the Manhattan road network and New York City yellow taxi data to simulate the real-time vehicle dispatching environment. Both centralized and decentralized taxi dispatching policies are examined with the simulator. This case study shows that the proposed approach can further improve taxi drivers' profits while reducing customers' waiting times compared to several existing vehicle dispatching algorithms.