Table of Content

30 September 2026, Volume 14 Issue 3

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Preface: Special Issue on AI for Optimization

Zhi-Ping Chen, Yu-Hong Dai, Tian-De Guo, Xin-Min Yang

2026, 14(3):  749-752.  doi:10.1007/s40305-026-00716-1

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A Universal Framework for Vehicle Routing Problems with Large Language Models

Rui-Bin Zeng, Mo-Xuan Yang, Ming-Long Lei, Ling-Feng Niu, Yu-Hong Dai

2026, 14(3):  753-783.  doi:10.1007/s40305-026-00692-6

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Vehicle routing problems (VRP) are typical NP-hard combinatorial optimization problems that have great significance in real-world applications. Recently, neural methods have shown promising abilities to solve VRP with superior quality while maintaining high efficiency. Existing neural methods still have limitations in terms of data requirements, especially for VRP variants. Furthermore, it is difficult to generalize across different types of VRPs with a unified learning framework without extensive retraining. Notably, the large language models (LLMs) have shown potential to assist in solving VRP problems. Therefore, this paper proposes a universal framework for VRP based on LLMs. This framework captures the essential characteristics of different VRPs in a unified manner and adapts to various problem specifications with less data reliance. Concretely, the core architecture is built on pre-training and fine-tuning. Low-rank adaptation and the mixture of experts are further integrated to improve the adaptation ability. Experimental results demonstrate that on multiple vehicle routing problems, such as the traveling salesman problem and the capacitated vehicle routing problem, our framework achieves promising solution quality and computational efficiency.

Contrastive Learning Enhanced Knowledge Distillation for Learning Distributionally Robust Vehicle Routing Problems

Rui-Yang Shi, Ling-Feng Niu, Xin Shen, Yu-Hong Dai

2026, 14(3):  784-805.  doi:10.1007/s40305-025-00632-w

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In recent years, learning based methods (LBMs) show great potential for solving vehicle routing problems (VRPs). However, most existing LBMs exhibit poor ability in distribution generalization,which hampers their applications in the real world.One reason behind this weakness is their underlying reinforcement learning training objective, which learns the best policy through interactions with the environment while ignoring the structural properties of training instances. To address this weakness and improve the distributional robustness of learned solvers, we propose a novel framework that enhances the knowledge distillation scheme with self-supervised contrastive learning. Compared with reinforcement learning, self-supervised learning can extract rich information from inherent feature representations and relationships of data. Specifically, we introduce a lightweight auxiliary network to the teacher network, so that the corresponding self-supervised signals can be extracted from this auxiliary module for knowledge distillation. By measuring the similarity between self-supervised signals and exploiting it to guide the training process, useful hidden information can be effectively transferred from the teacher network to the student network. Numerical experiments demonstrate the effectiveness of our proposed method, especially in improving the models’ generalization ability.

Neural Re-Insertion Improvement Algorithm for Large-Scale TSP

Wen-Zhao Liu, Cong-Ying Han, Tian-De Guo

2026, 14(3):  806-832.  doi:10.1007/s40305-025-00669-x

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Traveling salesman problem (TSP) is an important problem in combinatorial optimization, which serves as the foundation for significant frontier scientific issues, including integrated circuit design and genomics. Learning-based methods have shown impressive results in solving the TSP, demonstrating efficient perception and decision-making abilities. However, as the scale of modern scenarios continues to expand, efficiently improving existing solutions through learning approaches remains a challenging and urgent task. A general framework for re-insertion improvement in the TSP is proposed. Focusing the core decision on the selection of re-insertion positions, we design a neural re-insertion improvement (NRII) algorithm based on deep reinforcement learning. NRII integrates multiple attention mechanisms and neural components to achieve the perception and parameterization of re-insertion positions, followed by model training through a policy optimization method. Extensive numerical experiments validate that the NRII algorithm can be combined with various solving methods to efficiently improve the solution quality across different scales and distributions. Additionally, detailed sensitivity analyses of the hyperparameter in NRII are also conducted.

ARPO: Optimal Robust Policy Optimization Models from Intrinsic State-Adversarial Markov Decision Process

Jia-Yu Lv, Hao-Ran Li, Cong-Ying Han, Tian-De Guo

2026, 14(3):  833-859.  doi:10.1007/s40305-026-00682-8

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Reinforcement learning (RL) faces significant challenges when deployed in real-world scenarios with potential adversarial perturbations. The Intrinsic State-Adversarial Markov Decision Process (ISA-MDP) framework considers state perturbation to enhance policy robustness while preserving its natural performance. Building upon ISA-MDP, this paper proposes the first policy optimization framework and develops two distinct algorithms for different scenarios: Deterministic Adversarially Robust Policy Gradient (DARPG) and stochastic adversarially robust policy optimization, which effectively solve our formulated optimization problem. Furthermore,we provide a convergence analysis for both algorithms under specified theoretical assumptions.

Optimal Bailout for Systemic Risk: A PGO Approach Based on Neural Network

Shu-Hua Xiao, Jia-Li Ma, Li Xia, Shu-Shang Zhu

2026, 14(3):  860-890.  doi:10.1007/s40305-025-00663-3

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In the financial system, bailout strategies play a pivotal role in mitigating substantial losses resulting from systemic risk. However, the lack of a closed-form objective function to the optimal bailout problem poses significant challenges in its resolution. This paper conceptualizes the optimal bailout (capital injection) problem as a black-box optimization task, where the black box is modeled as a fixed-point system consistent with the E–N framework for measuring systemic risk in the financial system. To address this challenge, we propose a novel framework, “Prediction-Gradient-Optimization” (PGO). Within PGO, the Prediction employs a neural network to approximate and forecast the objective function implied by the black box, which can be completed offline; for the online usage, the Gradient step derives gradient information from this approximation, and the Optimization step uses a gradient projection algorithm to solve the problem effectively. Extensive numerical experiments highlight the effectiveness of the proposed approach in managing systemic risk.

An RML-Based Asymmetric Loss Support Vector Machine for Robust and Imbalanced Classification

Yu-Nuo Chen, Kai Qi, Li-Ping Tang

2026, 14(3):  891-945.  doi:10.1007/s40305-025-00662-4

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Support vector machines (SVMs) are powerful supervised learning methods widely used for classification and regression. However, theirs performance can significantly deteriorate in the presence of label noise and class imbalance, which are common in real-world applications. To address these issues, we propose a novel asymmetric bounded loss function within the general framework of robust loss functions for machine learning (RML). This loss function allows for class-dependent misclassification penalties while ensuring robustness to label noise. By incorporating this loss into the SVM framework, we develop a new model, referred to as asymmetric loss SVM (ALSVM), which is efficiently optimized via the stochastic variance reduced gradient method. We further provide a theoretical analysis by deriving generalization error bounds based on Rademacher complexity, and demonstrate the robustness by showing that its influence function is bounded. Extensive experiments on benchmark datasets, particularly those related to disease classification, demonstrate that the proposed method consistently improves performance across multiple evaluation metrics.

A Dual SVM Approach to Noisy Labels Relabeling

Ali Sahleh, Maziar Salahi, Sadegh Eskandari

2026, 14(3):  946-963.  doi:10.1007/s40305-024-00575-8

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In machine learning, models are learned from training data with labels denoting the assigned names or classes and features representing characteristics of samples. However, data collection may introduce noise which can significantly impact the performance of these algorithms. In this paper, to take benefits of the computational efficiency of dual support vector machine (SVM) model, first we implement relabeling on the dual SVM problem resulting in a mixed-integer program (MIP) with less variables and constraints compared to the primal relabeled SVM (RESVM) model in the literature. Then we propose another dual SVM-based MIP model called cluster dual RESVM (CDRESVM), wherein instances are relabeled based on their proximity to the nearest class by using binary variables and new set of constraints. Experiments on 10 UCI and 2 Microarray datasets show that the proposed dual SVM-based models exhibit higher accuracy and better learning time compared to the primal SVM models, on the majority of datasets.

Stochastic Functions Learning from Distribution-Driven Data: Generalization Bound and Algorithms

Jia Wu, Xian-Tao Xiao, Li-Wei Zhang

2026, 14(3):  964-994.  doi:10.1007/s40305-024-00563-y

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This paper delves into the learning of stochastic functions from distribution-driven data. Utilizing the empirical risk minimization (ERM) approach, we construct a stochastic minimax optimization problem from the distribution-driven data to recover the target random function. The generalization bound of the ERM approach is established based on annealed VC entropy. When the hypothesis set of functions is parameterized by a vector over a compact set, the optimization problem is simplified to a finite-dimensional stochastic composite minimax problem. This problem can be solved through a stochastic compositional gradient descent ascent method. Under mild conditions, the method is proved to converge almost surely to an optimal solution and achieve a sublinear convergence rate.

SOR-Unrolled Learning-to-Solve Method for Absolute Value Equations

Cheng-Xi Song, Zi-Yang Zhang, Rui-Zhi Zhou, Ling-Feng Niu, Yu-Hong Dai

2026, 14(3):  995-1013.  doi:10.1007/s40305-025-00670-4

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The absolute value equation (AVE), which is equivalent to the linear complementarity problem and mixed integer programming, has wide applications. Determining a solution to the AVE is NP-hard, and many algorithms have been proposed. SOR-like iteration, which is designed based on fixed point iteration and has a simple form and good theoretical guarantee, is one of the state-of-the-art methods for solving the AVE. However, its performance heavily relies on complex parameter selection. In this paper, we utilize deep neural networks to avoid difficult parameter selection. Concretely, we propose the first deep unrolled neural network based on SOR-like iteration to solve the AVE. On the one hand, in theory, we verify the feasibility of approximating the solution to AVE by unrolling SOR-like iterations with neural networks. On the other hand, in practice, we propose a specialized network architecture called SOR-Net, which incorporates learnable parameters and nonlinear activation functions into a deep network, with computational layers unrolled from the SOR-like iterations. Furthermore, to efficiently obtain more accurate solutions and further leverage the strengths of neural networks, we design a two-stage framework. The first stage treats the sign prediction in the AVE as a binary classification task using SOR-Net, effectively reducing the original AVE to a standard linear equation system. Then, in the second stage, this system can be solved using traditional numerical algorithms with higher precision. Extensive numerical experiments validate the effectiveness and efficiency of our proposed approach.

Global Convergence of a Modified BFGS-Type Method Based on Function Information for Nonconvex Multiobjective Optimization Problems

Ying-Xue Yang, Xin Deng, Li-Ping Tang

2026, 14(3):  1014-1037.  doi:10.1007/s40305-025-00589-w

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Most of the quasi-Newton methods for multiobjective optimization require independent approximations of the Hessian matrix for each objective function at each iteration, which leads to significant storage and computational burdens. To address this issue, in this paper, we propose a novel modified BFGS-type method for nonconvex multiobjective optimization problems (M-BFGSMO) based on function information. The M-BFGSMO framework utilizes a shared BFGS-type matrix to approximate the Hessian matrix of all objective functions simultaneously, updating it iteratively using gradient and function value information from previous iterations. This approach balances computational efficiency with algorithmic performance. We establish the convergence of the M-BFGSMO method without the requirement of convexity, and prove its R-linear convergence under mild conditions. Numerical experiments on test problems, including both convex and nonconvex cases, demonstrate the effectiveness and computational advantages of the proposed method compared to some existing approaches.

Self-adaptive Technique with Double Inertial Steps for Inclusion Problem on Hadamard Manifolds

Hammed Anuoluwapo Abass, Olawale Kazeem Oyewole, Kazeem Olalekan Aremu, Lateef Olakunle Jolaoso

2026, 14(3):  1038-1060.  doi:10.1007/s40305-024-00537-0

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In this article, we investigate monotone and Lipschitz continuous variational inclusion problem in the settings of Hadamard manifolds. We propose a forward–backward method with a self-adaptive technique for solving variational inclusion problem. To increase the rate of convergence of our proposed method, we incorporate our iterative method with double inertial steps and establish a convergence result of our iterative method under some mild conditions. Finally, in order to illustrate the computational effectiveness of our method, some numerical examples are also discussed. The result present in this article is new in this space and extends many related results in the literature.

Algorithms for Finding Minimum-Norm Solutions of Quasimonotone Variational Inequalities

Duong Viet Thong, Xiao-Huan Li, Pham Thi Hong Tham, Nguyen Hai Duong

2026, 14(3):  1061-1087.  doi:10.1007/s40305-025-00606-y

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This paper introduces two novel self-adaptive extragradient projection methods for solving variational inequalities involving Lipschitz continuous and quasimonotone mappings in real Hilbert spaces. Our proposed methods incorporate an inertial step into a single projection framework, leveraging the potential advantages of inertial techniques to accelerate convergence. We rigorously establish strong convergence theorems for the proposed algorithms, providing theoretical guarantees for their convergence behavior under appropriate assumptions. Furthermore, to evaluate the practical efficacy of our methods, we present comprehensive numerical experiments and comparisons.

Alternating Bayesian Mixed Integer Optimization for Context Window Extension in Large Language Models Without Tuning Model Parameters

Jia Liu, Zhi-You Wu, Huan Gao, Peng Hu, Ran Gu

2026, 14(3):  1088-1114.  doi:10.1007/s40305-026-00672-w

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Extending the context window of large language models (LLMs) is crucial for effectively processing long texts, yet existing models typically struggle due to their restricted input lengths. Existing methods for extending context windows mainly include training-based and training-free approaches. However, training-based methods require significant computational resources, while training-free methods typically achieve limited performance. In this paper, we propose a new method to effectively extend the context window of LLMs. This method consists of three key components: First, we conduct an experimental analysis of Rotary Position Embedding (RoPE), a primary positional encoding method used in LLMs, to identify the critical characteristics that influence the context window size and performance of LLMs. Second, based on the analysis results, we formulate a mixed integer black-box optimization model and utilize its optimal solution to determine the best adjustment strategy for RoPE. Third, we develop an alternating Bayesian mixed integer optimization algorithm to solve the model. Extensive experiments show that our method successfully extends the context window of LLMs and outperforms most existing training-based and training-free methods across various tasks.

An Inertial Extragradient Method for Solving Variational Inequalities Without Monotonicity

Duong Viet Thong, Xiao-Huan Li, Hoang Van Thang, Luong Van Long

2026, 14(3):  1115-1139.  doi:10.1007/s40305-025-00665-1

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In this work, we introduce an extragradient method with one-step inertial and self-adaptive step sizes to solve variational inequalities. We obtained weak convergence of the generated sequence under some appropriate conditions on both the iterative parameters and operator. In particular, our algorithm uses new explicit step sizes which plays an important role in improving the speed of the algorithm and weak convergence is obtained without using the monotonicity of the cost operator with one projection onto the feasible set and two evaluations of the operator per iteration. Finally, some numerical results show that our method is efficient and competitive with other methods in the literature.