Automated Heuristic Design (AHD) using Large Language Models (LLMs) has achieved notable success in the past two years. Despite the effectiveness of existing approaches, they only design a single heuristic to serve all problem instances, often inducing poor generalization across different distributions or sizes. To address this issue, we propose Automated Heuristic Set Design (AHSD), a new methodology for LLM-driven AHD. The aim of AHSD is to automatically design a small-sized complementary heuristic set to serve diverse problem instances, such that each problem instance could be optimized by at least one heuristic in this set. We propose Evolution of Heuristic Set (EoH-S), which realizes AHSD using an evolutionary search framework. It incorporates a complementary population management and a memetic search to design a set of heuristics. Extensive experiments on online bin packing, traveling salesman problem, and capacitated vehicle routing problem show that EoH-S consistently outperforms existing AHD methods. The resulting heuristics exhibit complementary performance across instances of varying sizes and distributions.

Fluid antenna is a new reconfigurable antenna technology that can dynamically adjust the positions or ports of radiating elements and therefore provides a new degree of freedom for wireless communications. However, the associated port selection is a challenging large-scale combinatorial optimization problem and difficult to solve. Existing manually designed heuristic algorithms are not only labor-intensive, but cannot achieve satisfactory performance. In this paper, we propose a novel paradigm that leverages large language models (LLMs) for automated design of optimization algorithms for fluid antenna systems without manual hyperheuristic tuning. Specifically, we study the problem of maximizing the minimum signal-to-interference-plus-noise ratio (SINR) in the downlink to ensure fairness among users by optimizing port selection and beamforming. We investigate two LLM-enabled algorithm optimization strategies. The first is to optimize the crossover and mutation operations to enhance the performance of the well-known genetic algorithm and the second is to design AutoPort, a new heuristic from scratch by LLM, to solve the optimization problem. Simulation results verify that the proposed method can achieve near-optimal performance and significant improvement over the conventional genetic algorithm and the deep learning approach.

Recent advancements in Neural Combinatorial Optimization (NCO) have shown promise in solving routing problems like the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) without handcrafted designs. Research in this domain has explored two primary categories of methods: iterative and non-iterative. While non-iterative methods struggle to generate near-optimal solutions directly, iterative methods simplify the task by learning local search steps. However, existing iterative methods are often limited by restricted neighborhood searches, leading to suboptimal results. To address this limitation, we propose a novel approach that extends the search to larger neighborhoods by learning a destroy-and-repair strategy. Specifically, we introduce a Destroy-and-Repair framework based on Hyper-Graphs (DRHG). This framework reduces consecutive intact edges to hyper-edges, allowing the model to pay more attention to the destroyed part and decrease the complexity of encoding all nodes. Experiments demonstrate that DRHG achieves state-of-the-art performance on TSP with up to 10,000 nodes and shows strong generalization to real-world TSPLib and CVRPLib problems.

Heuristics are commonly used to tackle various search and optimization problems. Design heuristics usually require tedious manual crafting with domain knowledge. Recent works have incorporated Large Language Models (LLMs) into automatic heuristic search, leveraging their powerful language and coding capacity. However, existing research focuses on the optimal performance on the target problem as the sole objective, neglecting other criteria such as efficiency and scalability, which are vital in practice. To tackle this challenge, we propose to model the heuristic search as a multi-objective optimization problem and consider introducing additional practical criteria beyond optimal performance. Due to the complexity of the search space, conventional multi-objective optimization methods struggle to effectively handle LLM-based multi-objective heuristic search. We propose the first LLM-based multi-objective heuristic search framework, Multi-objective Evolution of Heuristic (MEoH), which integrates LLMs in a zero-shot manner to generate a non-dominated set of heuristics to meet multiple design criteria. We design a new dominance-dissimilarity mechanism for effective population management and selection, which incorporates both code dissimilarity in the search space and dominance in the objective space. MEoH is demonstrated in two well-known combinatorial optimization problems: the online Bin Packing Problem (BPP) and the Traveling Salesman Problem (TSP). The results indicate that a variety of elite heuristics are automatically generated in a single run, offering more trade-off options than the existing methods. It successfully achieves competitive or superior performance while improving efficiency up to 10 times. Moreover, we also observe that the multi-objective search introduces novel insights into heuristic design and leads to the discovery of diverse heuristics.

Handcrafting heuristics often demands extensive domain knowledge and significant development effort. Recently, heuristic search powered by large language models (LLMs) has emerged as a new approach, offering enhanced automation and promising performance. Existing methods rely on an evolutionary computation (EC) framework with carefully designed prompt strategies. However, the large heuristic search space poses significant challenges for these EC-based methods. This paper proposes a simple yet effective LLM-driven Heuristic Neighborhood Search (LHNS) paradigm to iteratively search in the heuristic neighborhood in a principled way for efficient heuristic design. Three distinct methods are designed under this neighborhood search paradigm and demonstrated on three widely studied problems. Results indicate that LHNS exhibits very competitive performance and surpasses existing EC-based methods in efficiency. It also demonstrates sufficient robustness in the absence of problem-specific knowledge regarding the target problem. The efficiency and robust adaptability make it a practical new solution for efficient heuristic design.

Crafting adversarial examples is crucial for evaluating and enhancing the robustness of Deep Neural Networks (DNNs), presenting a challenge equivalent to maximizing a non-differentiable 0-1 loss function. However, existing single objective methods, namely adversarial attacks focus on a surrogate loss function, do not fully harness the benefits of engaging multiple loss functions, as a result of insufficient understanding of their synergistic and conflicting nature. To overcome these limitations, we propose the Multi-Objective Set-Based Attack (MOS Attack), a novel adversarial attack framework leveraging multiple loss functions and automatically uncovering their interrelations. The MOS Attack adopts a set-based multi-objective optimization strategy, enabling the incorporation of numerous loss functions without additional parameters. It also automatically mines synergistic patterns among various losses, facilitating the generation of potent adversarial attacks with fewer objectives. Extensive experiments have shown that our MOS Attack outperforms single-objective attacks. Furthermore, by harnessing the identified synergistic patterns, MOS Attack continues to show superior results with a reduced number of loss functions. Our code is available at https://github.com/pgg3/MOS-Attack.

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.

Vehicle routing problems (VRPs) are significant combinatorial optimization problems (COPs) holding substantial practical importance. Recently, neural combinatorial optimization (NCO), which involves training deep learning models on extensive data to learn vehicle routing heuristics, has emerged as a promising approach due to its efficiency and the reduced need for manual algorithm design. However, applying NCO across diverse real-world scenarios with various constraints necessitates cross-problem capabilities. Current cross-problem NCO methods for VRPs typically employ a constraint-unaware model, limiting their cross-problem performance. Furthermore, they rely solely on global connectivity, which fails to focus on key nodes and leads to inefficient representation learning. This paper introduces a Constraint-Aware Dual-Attention Model (CaDA), designed to address these limitations. CaDA incorporates a constraint prompt that efficiently represents different problem variants. Additionally, it features a dual-attention mechanism with a global branch for capturing broader graph-wide information and a sparse branch that selectively focuses on the key node connections. We comprehensively evaluate our model on 16 different VRPs and compare its performance against existing cross-problem VRP solvers. CaDA achieves state-of-the-art results across all tested VRPs. Our ablation study confirms that each component contributes to its cross-problem learning performance. The source code for CaDA is publicly available at https://github.com/CIAM-Group/CaDA.

Causal structure learning (CSL) plays a pivotal role in causality and is often formulated as an optimization problem within score-and-search methods. Under the assumption of an infinite dataset and a predefined distribution, several well-established and consistent score functions have been shown to be both optimal and reliable for identifying ground-truth causal graphs. However, in practice, these idealized assumptions are often infeasible, which can result in CSL algorithms learning suboptimal structures. In this paper, we introduce L-SFE, a framework designed to automatically discover effective score functions by exploring the "score function space". L-SFE addresses this task from a bi-level optimization perspective. First, it leverages a Large Language Model (LLM) to interpret the characteristics of score functions and generate the corresponding code implementations. Next, L-SFE employs evolutionary algorithms along with carefully designed operators, to search for solutions with higher fitness. Additionally, we take the BIC as example and prove the consistency of the generated score functions. Experimental evaluations, conducted on discrete, continuous, and real datasets, demonstrate the high stability, generality and effectiveness of L-SFE.

Combinatorial optimization (CO) is fundamental to several realworld applications, from logistics and scheduling to hardware design and resource allocation.Deep reinforcement learning (RL) has recently shown significant benefits in solving CO problems, reducing reliance on domain expertise and improving computational efficiency.However, the absence of a unified benchmarking framework leads to inconsistent evaluations, limits reproducibility, and increases engineering overhead, raising barriers to adoption for new researchers.To address these challenges, we introduce RL4CO, a unified and extensive benchmark with in-depth library coverage of 27 CO problem environments and 23 state-of-the-art baselines.Built on efficient software libraries and best practices in implementation, RL4CO features modularized implementation and flexible configurations of diverse environments, policy architectures, RL algorithms, and utilities with extensive documentation.RL4CO helps researchers build on existing successes while exploring and developing their own designs, facilitating the entire research process by decoupling science from heavy engineering.We finally provide extensive benchmark studies to inspire new insights and future work.RL4CO has already attracted numerous researchers in the community and is open-sourced at https://github.com/ai4co/rl4co 1 .

Neural Combinatorial Optimisation (NCO) is a promising learning-based approach for solving Vehicle Routing Problems (VRPs) without extensive manual design. While existing constructive NCO methods typically follow an appending-based paradigm that sequentially adds unvisited nodes to partial solutions, this rigid approach often leads to suboptimal results. To overcome this limitation, we explore the idea of the insertion-based paradigm and propose Learning to Construct with Insertion-based Paradigm (L2C-Insert), a novel learning-based method for constructive NCO. Unlike traditional approaches, L2C-Insert builds solutions by strategically inserting unvisited nodes at any valid position in the current partial solution, which can significantly enhance the flexibility and solution quality. The proposed framework introduces three key components: a novel model architecture for precise insertion position prediction, an efficient training scheme for model optimization, and an advanced inference technique that fully exploits the insertion paradigm's flexibility. Extensive experiments on both synthetic and real-world instances of the Travelling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that L2C-Insert consistently achieves superior performance across various problem sizes. The code is available at [https://github.com/CIAM-Group/L2C\_Insert](https://github.com/CIAM-Group/L2C\_Insert).

This paper studies a multiobjective bandit problem under lexicographic ordering, wherein the learner aims to maximize $m$ objectives, each with different levels of importance. First, we introduce the local trade-off, $\lambda_*$, which depicts the trade-off between different objectives. For the case when an upper bound of $\lambda_*$ is known, i.e., $\lambda\geq\lambda_*$, we develop an algorithm that achieves a general regret bound of $\widetilde{O}(\Lambda^i(\lambda)T^{(d_z^i+1)/(d_z^i+2)})$ for the $i$-th objective, where $i\in\{1,2,\ldots,m\}$, $\Lambda^i(\lambda)=1+\lambda+\cdots+\lambda^{i-1}$, $d_z^i$ is the zooming dimension for the $i$-th objective, and $T$ is the time horizon. Next, we provide a matching lower bound for the lexicographic Lipschitz bandit problem, proving that our algorithm is optimal in terms of $\lambda_*$ and $T$. Finally, for the case where $m=2$, we remove the dependence on the knowledge about $\lambda_*$, albeit at the cost of increasing the regret bound to $\widetilde{O}(\Lambda^i(\lambda_*)T^{(3d_z^i+4)/(3d_z^i+6)})$, which remains optimal in terms of $\lambda_*$. Compared to existing work on lexicographic multi-armed bandits, our approach improves the current regret bound of $\widetilde{O}(T^{2/3})$ and extends the number of arms to infinity. Numerical experiments confirm the effectiveness of our algorithms.

Logistics transportation plays a critical role in real-world applications. The heterogeneous fleet vehicle routing problem (HFVRP), characterized by varying vehicle capacities and costs, are the key optimization challenges in many logistic scenarios. Despite its importance, it presents substantial challenges due to its NP-hard nature and large scale. Existing methods only study HFVRP instances of moderate size (i.e., about 300 nodes), which is insufficient for real-world application. In this article, we introduce large language model-multipopulation (MP-LLM), a novel MP optimization method with LLM-driven knowledge discovery. MP-LLM employs multiple populations with iterated local search (ILS) and dynamic updating to balance exploration and exploitation. An LLM-driven knowledge discovery is adopted to design a parameter adjustment strategy to pinpoint features specific to each instance, thereby facilitating a more effective dynamic parameter adjustment. We comprehensively evaluate MP-LLM on four benchmark test sets with 170 instances of diverse distributions and sizes. Our results show that when compared to stat-of-the-art methods, MP-LLM not only achieves superior solution quality but also significantly enhances efficiency. Notably, MP-LLM generates new best-known solutions on 18 out of 90 classic instances. It significantly expands HFVRP-solving capabilities from approximately 300 nodes to instances with up to 3000 nodes.

Many combinatorial multiobjective optimization problems involve very costly-to-evaluate objectives and constraints. It is very difficult, if not impossible, for traditional heuristics to solve these problems with an acceptable amount of computational time. In this paper, we show that offline machine learning can be very useful to assist multiobjective evolutionary algorithms to tackle this kind of problem. We take a complicated real-life multiobjective routing-packing problem as the test bed. We propose to use offline machine learning methods to replace time-consuming packing heuristics for packing feasibility prediction. Experiments show that the machine learning models can be 1,000 times faster than some commonly used packing heuristics and their accuracy can be as high as 98%. We adopt MOEA/D to decompose the problem into a number of single objective subproblems and solve them in a collaborative manner. We propose an encoding strategy to represent each routing scheme and use genetic operators to generate new routes. Experimental studies have been conducted on 100 instances from HUAWEI’s real-world logistics application and two test suites from the literature. Our proposed method can solve each HUAWEI instance in around one minute. Our solutions on the two test suites are comparable to other existing algorithms, and the overall computational cost of our method is significantly lower than others.

This paper studies the multiobjective bandit problem under lexicographic ordering, wherein the learner aims to simultaneously maximize ? objectives hierarchically. The only existing algorithm for this problem considers the multi-armed bandit model, and its regret bound is O((KT)^(2/3)) under a metric called priority-based regret. However, this bound is suboptimal, as the lower bound for single objective multi-armed bandits is Omega(KlogT). Moreover, this bound becomes vacuous when the arm number K is infinite. To address these limitations, we investigate the multiobjective Lipschitz bandit model, which allows for an infinite arm set. Utilizing a newly designed multi-stage decision-making strategy, we develop an improved algorithm that achieves a general regret bound of O(T^((d_z^i+1)/(d_z^i+2))) for the i-th objective, where d_z^i is the zooming dimension for the i-th objective, with i in {1,2,...,m}. This bound matches the lower bound of the single objective Lipschitz bandit problem in terms of T, indicating that our algorithm is almost optimal. Numerical experiments confirm the effectiveness of our algorithm.

When gradient information is available, multi-objective Bayesian optimization (MOBO) algorithm using gradient-enhanced Gaussian process (GradGP) models has been proven to be an efficient method. However, the intractable training and inference complexity hinders its scalability in high-throughput scenarios with large batch sizes and increasing data volume. To address this issue, we develop an efficient high-throughput MOBO algorithm using gradient information named MOEAID-DSVGP, which integrates stochastic variational GP models using directional derivatives (DSVGP) within the decomposition-based MOBO framework. In this algorithm, DSVGP model is trained for each objective function by optimizing a small set of inducing inputs and inducing directions as variational parameters, the complexity is independent of sample size and input dimension. MOEAID is employed as the inner optimizer for high-throughput solution generation. Experimental studies with varying batch sizes and input dimensions show that compared to GradGP-based MOBO method, the proposed algorithm exhibits superior scalability and allows more optimization iterations to yield better outcome in most cases. We further investigate and analyze the influence of increasing the number of inducing directions on the performance improvement in problems with different features.

Multi-objective optimization problems can be found in many real-world applications, where the objectives often conflict each other and cannot be optimized by a single solution. In the past few decades, numerous methods have been proposed to find Pareto solutions that represent optimal trade-offs among the objectives for a given problem. However, these existing methods could have high computational complexity or may not have good theoretical properties for solving a general differentiable multi-objective optimization problem. In this work, by leveraging the smooth optimization technique, we propose a lightweight and efficient smooth Tchebycheff scalarization approach for gradient-based multi-objective optimization. It has good theoretical properties for finding all Pareto solutions with valid trade-off preferences, while enjoying significantly lower computational complexity compared to other methods. Experimental results on various real-world application problems fully demonstrate the effectiveness of our proposed method.

Heuristics are widely used for dealing with complex search and optimization problems. However, manual design of heuristics can be often very labour extensive and requires rich working experience and knowledge. This paper proposes Evolution of Heuristic (EoH), a novel evolutionary paradigm that leverages both Large Language Models (LLMs) and Evolutionary Computation (EC) methods for Automatic Heuristic Design (AHD). EoH represents the ideas of heuristics in natural language, termed thoughts. They are then translated into executable codes by LLMs. The evolution of both thoughts and codes in an evolutionary search framework makes it very effective and efficient for generating high-performance heuristics. Experiments on three widely studied combinatorial optimization benchmark problems demonstrate that EoH outperforms commonly used handcrafted heuristics and other recent AHD methods including FunSearch. Particularly, the heuristic produced by EoH with a low computational budget (in terms of the number of queries to LLMs) significantly outperforms widely-used human hand-crafted baseline algorithms for the online bin packing problem.

Vehicle routing problems (VRP) are very important in many realworld applications and has been studied for several decades.Recently, neural combinatorial optimization (NCO) has attracted growing research effort.NCO is to train a neural network model to solve an optimization problem in question.However, existing NCO methods often build a different model for each routing problem, which significantly hinders their application in some areas where there are many different VRP variants to solve.In this work, we make a first attempt to tackle the crucial challenge of cross-problem generalization in NCO.We formulate VRPs as different combinations of a set of shared underlying attributes and solve them simultaneously via a single model through attribute composition.In this way, our proposed model can successfully solve VRPs with unseen attribute combinations in a zero-shot generalization manner.In our experiments, the neural model is trained on five VRP variants and its performance is tested on eleven VRP variants.The experimental results show that the model demonstrates superior performance on these eleven VRP variants, reducing the average gap to around 5% from over 20% and achieving a notable performance boost on both benchmark datasets and real-world logistics scenarios.

The Vehicle routing problems (VRP) are one of the most studied combinatorial optimization problems. This paper targets an important and challenging VRP variant, named multi-objective vehicle routing problems with time windows. We propose to use a multi-objective evolutionary algorithm based on decomposition (MOEA/D) to decompose the problem into a set of single-objective sub-problems. For each sub-problem, efficient crossover and local search heuristics are adopted to generate and improve new solutions. We design two new strategies to decrease the number of vehicles, a population management method with infeasible solutions and a multi-split strategy. Experimental studies are carried out on the well-known Solomon's dataset. Results suggest that our proposed algorithm is very competitive compared to two state-of-the-art algorithms. It generates better solutions on 90% of the test instances.

Neural combinatorial optimization (NCO) is a promising learning-based approach for solving challenging combinatorial optimization problems without specialized algorithm design by experts. However, most constructive NCO methods cannot solve problems with large-scale instance sizes, which significantly diminishes their usefulness for real-world applications. In this work, we propose a novel Light Encoder and Heavy Decoder (LEHD) model with a strong generalization ability to address this critical issue. The LEHD model can learn to dynamically capture the relationships between all available nodes of varying sizes, which is beneficial for model generalization to problems of various scales. Moreover, we develop a data-efficient training scheme and a flexible solution construction mechanism for the proposed LEHD model. By training on small-scale problem instances, the LEHD model can generate nearly optimal solutions for the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 1000 nodes, and also generalizes well to solve real-world TSPLib and CVRPLib problems. These results confirm our proposed LEHD model can significantly improve the state-of-the-art performance for constructive NCO.

For real-world engineering design optimizations, it is of great significance to find approximate optimal designs with least number of expensive functional evaluations. This paper proposes to use a surrogate-based multi-objective evolutionary algorithm (SBMO) to address this type of problems. The basic idea is to decompose a multi-objective optimization problem into a number of scalar optimization subproblems and to optimize them simultaneously in a simple-to-implement manner, in which global surrogate models are used to enable full cooperation between subproblems. First, initial samples are selected by design of experiments and expensive simulations are conducted to evaluate them. Second, global surrogate models for objective (and constraint) functions are built through the sampled data and the optimization subproblems are solved simultaneously to suggest new samples. Third, the surrogate models are updated and the optimization proceeds to the next generation. This process is repeated until satisfactory Pareto-front solutions are found. Thanks to decomposition strategy, the infill-sampling criteria and constraint handling dedicated for a single-objective optimization can be directly used in a SBMO. The difference between SBMO and the existing methods such as MOEA/D-EGO is that a combined infill-sampling strategy and dedicated constraint handling are used. Benchmark test cases have demonstrated that SBMO is efficient, robust and has good capability of constraint handling. SBMO has been applied to multi-objective aerodynamic shape optimization of a transonic airfoil. It has been shown that SBMO is well suited for engineering design problems where expensive numerical simulations are employed.