The Heterogeneous Fleet Vehicle Routing Problem (HFVRP) aims to find optimal routes for vehicles with different capacities and costs, and is common in real-world applications. Total cost and fairness among drivers are two important yet conflicting objectives, while existing studies address either one objective alone or a specific weighted sum of them. To trade off the two objectives simultaneously, this paper formulates the Multi-Objective HFVRP (MO-HFVRP). Our analysis reveals that the MO-HFVRP is challenging, as the decision space has sparse feasible solutions and the objective space exhibits an uneven distribution of objective vectors. Subsequently, a corresponding algorithm called AMOILS/D is proposed. It decomposes the MO-HFVRP into a few single-objective subproblems, and then applies Iterated Local Search (ILS) and multi-objective optimization techniques to collaboratively solve them. AMOILS/D has three key components. The first is the resource allocation strategy that periodically selects subproblems to focus the search on promising regions. The other two are the adaptive perturbation degree control and the acceptance mechanism in ILS. They enable effective navigation of the decision space and balance convergence and diversity. Experimental results show that AMOILS/D significantly outperforms other representative algorithms across most instances. Ablation studies also confirm the effectiveness of each proposed component.

Surrogate-assisted evolutionary algorithms (SAEAs) are well-suited for optimizing computationally expensive black-box problems in diverse real-world scenarios. The sample efficiency of SAEAs depends largely on the configuration of the surrogate model and sampling criteria. However, configuring these core components requires substantial manual effort and expert knowledge, limiting the broader applicability of SAEAs. To address these challenges, we propose CoE-SAEA, a novel paradigm that co-evolves large language models (LLMs) and configuration strategies to enhance SAEAs. Specifically, the paradigm consists of three populations with distinct roles: one evolves LLM prompts to generate robust configuration strategy instructions, another optimizes the configuration strategies, and the third solves the optimization problem using the selected algorithm configuration. Additionally, an exploration-exploitation module is incorporated to decide whether to explore new configuration strategies via LLMs or exploit existing ones. We empirically validate the efficacy of CoE-SAEA by comparing it to state-of-the-art algorithms across various benchmark problems and a real-world traffic signal optimization task. The source code of the proposed CoE-SAEA is publicly available at: https://github.com/ForrestXie9/CoE-SAEA.

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.

The coupling of multiple constraints can pose difficulties in solving constrained multi-objective optimization problems (CMOPs). Existing constrained multi-objective evolutionary algorithms (CMOEAs) often overlook this issue by considering all constraints together. This article proposes MTOTC, a novel multi-tasking optimization algorithm that addresses this challenge through a task clone technique. MTOTC clones the target CMOP with q constraints into q+1 copies, resulting in a total of q+2 tasks. Each cloned task is handled using an independent population that considers a unique constraint-handling sequence, effectively decoupling the constraints in q+1 different ways. Additionally, the algorithm incorporates online information sharing between the target task and cloned tasks, enabling the utilization of valuable search history as much as possible. Experimental results on four recently developed complex CMOP benchmark suites and a series of real-world CMOPs demonstrate the superior performance of MTOTC compared to seven state-of-the-art CMOEAs.

The hardly dominated boundary (HDB) is commonly observed in multi-objective optimization problems (HDBMOPs). However, there are only a few benchmark problems related to HDB-MOPs in the evolutionary computation community, which is insufficient to validate the performance of multi-objective evolutionary algorithms (MOEAs). In this paper, we first introduce a new set of HDB-MOPs characterized by various shapes of Pareto fronts and scalable HDB sizes. We then systematically analyze the capabilities of several representative existing MOEAs in handling HDB-MOPs and reveal their strengths and weaknesses in solving this type of problem. Finally, based on this insightful analysis, we propose an indicator-based MOEA with an adaptive reference point to effectively address HDB-MOPs. The source codes of the proposed benchmark problems and the IMOEAARP algorithm are available from https://github.com/CIAMGroup/ EvolutionaryAlgorithm Codes/tree/main/IMOEA-ARP.

Transforming a constrained optimization problem (COP) into a multi/many-objective optimization problem (MOP/MaOP) represents a practical approach for solving COPs. This article introduces an adaptive multi/many-objective transformation technique, termed adaptive many-objective transformation technique (AMaOTCO), designed to effectively address COPs. The transformed many-objective optimization problem (MaOP) defines an objective using a convex combination of the objective function (or constraint violation function) and an auxiliary function. This auxiliary function is constructed through a convex combination of the objective function and a weighted constraint violation function. The adaptive tuning of all combination coefficients is based on population information. This adaptive tuning ensures an intelligent balance between minimizing various constraint violations and managing the tradeoff between objective function minimization and constraint violation reduction. The effectiveness of the proposed AMaOTCO is demonstrated through comparisons with state-of-the-art constrained evolutionary algorithms (CEAs) on a set of real-world COPs.

Surrogate-assisted evolutionary algorithms for addressing expensive optimization problems with both continuous and categorical variables (EOPCCVs) are still in the early stages of development. This study makes significant advancements by leveraging the mixed-variable nature of EOPCCVs in two crucial ways. First, it introduces a novel hybrid approach combining differential evolution and upper confidence bound sampling (DEUCB), designed to explore the mixed search space effectively. Second, a specialized value distance metric (VDM) is proposed, integrating continuous and categorical variables, to enhance the accuracy of the radial basis function (RBF) model approximation. Finally, we present a customized evolutionary expensive optimization algorithm (CEEO), which seamlessly incorporates DEUCB and RBF-VDM into the widely utilized global and local surrogate-assisted evolutionary optimization framework. Experimental results, compared against state-of-the-art counterparts on three distinct sets of benchmark problems and a convolutional neural network hyperparameter optimization task, consistently affirm the efficacy of the proposed CEEO in addressing EOPCCVs. The source code for the proposed CEEO algorithm is available athttps://github.com/CIAM-Group/EvolutionaryAlgorithm_Codes/tree/main/CEEO_Code.

Multiobjective optimization (MOO) plays a critical role in various real-world domains. A major challenge therein is generating $K$ uniform Pareto-optimal solutions to represent the entire Pareto front. To address this issue, this paper firstly introduces \emph{fill distance} to evaluate the $K$ design points, which provides a quantitative metric for the representativeness of the design. However, directly specifying the optimal design that minimizes the fill distance is nearly intractable due to the nested $\min-\max-\min$ optimization problem. To address this, we propose a surrogate ``max-packing'' design for the fill distance design, which is easier to optimize and leads to a rate-optimal design with a fill distance at most $4\times$ the minimum value. Extensive experiments on synthetic and real-world benchmarks demonstrate that our proposed paradigm efficiently produces high-quality, representative solutions and outperforms baseline methods.

This article proposes utilizing a single deep reinforcement learning model to solve combinatorial multiobjective optimization problems. We use the well-known multiobjective traveling salesman problem (MOTSP) as an example. Our proposed method employs an encoder-decoder framework to learn the mapping from the MOTSP instance to its Pareto-optimal set. Specifically, it leverages a novel routing encoder to extract information for both the entire multiobjective aspect and every individual objective from the MOTSP instance. The global embeddings and each objective's embeddings are adaptively aggregated via a routing network to form the subproblems' embedding that can well represent the MOTSP features. Using a modified context embedding, the subproblems' embeddings are fed into a decoder to produce a set of approximate Pareto-optimal solutions in parallel. Additionally, we develop a Top-k baseline to enable more efficient data utilization and lightweight training for our proposed method. We compare our method with heuristic-based and learning-based ones on various types of MOTSP instances, and the experimental results show that our method can solve MOTSP instances in real-time and outperform the other algorithms, especially on large-scale problem instances.

Surrogate-assisted evolutionary algorithms (SAEAs) have proven to be effective in solving computationally expensive optimization problems (EOPs). However, the performance of SAEAs heavily relies on the surrogate model and infill criterion used. To improve the generalization of SAEAs and enable them to solve a wide range of EOPs, this paper proposes an SAEA called AutoSAEA, which features model and infill criterion auto-configuration. Specifically, AutoSAEA formulates model and infill criterion selection as a two-level multi-armed bandit problem (TL-MAB). The first and second levels cooperate in selecting the surrogate model and infill criterion, respectively. A two-level reward (TL-R) measures the value of the surrogate model and infill criterion, while a two-level upper confidence bound (TL-UCB) selects the model and infill criterion in an online manner. Numerous experiments validate the superiority of AutoSAEA over some state-of-the-art SAEAs on complex benchmark problems and a real-world oil reservoir production optimization problem.

As a research hotspot across logistics, operations research, and artificial intelligence, route planning has become a key technology for intelligent transportation systems. Recently, data-driven machine learning heuristics, including learning construction methods and learning improvement methods, have achieved remarkable success in solving single-objective route planning problems. However, many practical route planning scenarios must simultaneously consider multiple conflict objectives. For example, modern logistics companies often need to simultaneously minimize time budget, transportation cost, and vehicle pollution. Several learning construction methods are proposed for solving classical multi-objective route planning (MORP) problems, yet no learning improvement heuristics have been developed so far, even though they are acknowledged to be more efficient in narrowing the optimality gap. To fill this gap, this paper proposes a learning improvement MORP method, Pareto Improver (PI). PI employs a population-based mechanism to approximate the Pareto front with a single deep reinforcement learning model. The experimental results on various MORP problems show that PI can significantly outperform other state-of-the-art methods.

Few-shot relation extraction aims to recognize novel relations with few labeled sentences in each relation. Previous metric-based few-shot relation extraction algorithms identify relationships by comparing the prototypes generated by the few labeled sentences embedding with the embeddings of the query sentences using a trained metric function. However, as these domains always have considerable differences from those in the training dataset, the generalization ability of these approaches on unseen relations in many domains is limited. Since the prototype is necessary for obtaining relationships between entities in the latent space, we suggest learning more interpretable and efficient prototypes from prior knowledge and the intrinsic semantics of relations to extract new relations in various domains more effectively. By exploring the relationships between relations using prior information, we effectively improve the prototype representation of relations. By using contrastive learning to make the classification margins between sentence embedding more distinct, the prototype's geometric interpretability is enhanced. Additionally, utilizing a transfer learning approach for the cross-domain problem allows the generation process of the prototype to account for the gap between other domains, making the prototype more robust and enabling the better extraction of associations across multiple domains. The experiment results on the benchmark FewRel dataset demonstrate the advantages of the suggested method over some state-of-the-art approaches.

The objective reduction has been regarded as a basic issue in many-objective optimization. Existing objective reduction methods identify one set of essential objectives using an approximate nondominated front. However, if the Pareto front (PF) of a many-objective optimization problem (MaOP) is irregular, one single set of essential objectives may not be efficient for objective reduction. This article proposes to produce several different sets of essential objectives in objective reduction. More specifically, we use the Gaussian mixture model clustering to classify the obtained nondominated front into different subsets and perform objective reduction on each subset. Both an offline objective reduction method and an online objective reduction method are developed. The experimental results indicate that our proposed methods work well for MaOPs with degenerate or nondegenerate PFs.

The surrogate-assisted evolutionary algorithm (SAEA) is one of the most efficient approaches for solving expensive optimization problems. However, it still faces challenges when dealing with complex and high-dimensional problems. To fill this gap, a new algorithm (called SAMFEO) that combines surrogate-assisted and model-free evolutionary optimization is proposed in this article. SAMFEO consists of a local surrogate-assisted multioperator evolutionary optimization (LSA-MoEO) and a model-free single-operator evolutionary optimization (MF-SoEO). Specifically, LSA-MoEO adopts multiple evolutionary operators to generate a set of offspring and prescreens the best one as the final offspring by using a lightweight local surrogate model trained by some newest evaluated solutions. MF-SoEO follows the traditional evolutionary optimization paradigm and is triggered based on the optimization utility of the LSA-MoEO. It plays a crucial role in preventing the population from getting stagnation. Experimental results show that SAMFEO has significant advantages over several state-of-the-art SAEAs on some complex benchmark problems and one real-world problem.

This article proposes a three-level radial basis function (TLRBF)-assisted optimization algorithm for expensive optimization. It consists of three search procedures at each iteration: 1) the global exploration search is to find a solution by optimizing a global RBF approximation function subject to a distance constraint in the whole search space; 2) the subregion search is to generate a solution by minimizing an RBF approximation function in a subregion determined by fuzzy clustering; and 3) the local exploitation search is to generate a solution by solving a local RBF approximation model in the neighborhood of the current best solution. Compared with some other state-of-the-art algorithms on five commonly used scalable benchmark problems, ten CEC2015 computationally expensive problems, and a real-world airfoil design optimization problem, our proposed algorithm performs well for expensive optimization.

It is very difficult to locate multiple global optimal solutions (GOSs) of multimodal optimization problems (MMOPs). To deal with this issue, a penalty-based multimodal optimization differential evolution (DE), called PMODE, is developed in this article. In PMODE, a penalty strategy with a dynamic penalty radius is constructed to solve MMOPs. An elite selection mechanism is designed to identify and select elite solutions. The neighboring areas of these elite solutions are penalized. PMODE uses a popular DE variant-JADE as its search engine. The proposed PMODE is compared with several other state-of-the-art multimodal optimization algorithms on 20 MMOPs used in the IEEE CEC2013 special session. The experimental results show that PMODE performs better than other state-of-the-art methods.

This article introduces a special multitasking optimization problem (MTOP) called the competitive MTOP (CMTOP). Its distinctive characteristics are that all tasks’ objectives are comparable, and its optimal solution is the best one among the optimal solutions of all the individual problems. This article proposes an evolutionary algorithm with an online resource allocation strategy and an adaptive information transfer mechanism to solve the CMTOP. The experimental results on benchmark and real-world problems show that our proposed algorithm is effective and efficient.

This article proposes an evolutionary algorithm using multiple penalties and multiple local surrogates (MPMLS) for expensive constrained optimization. In each generation, MPMLS defines and optimizes a number of subproblems. Each subproblem penalizes the constraints in the original problem using a different penalty coefficient and has its own search subregion. A local surrogate is built for optimizing each subproblem. Two major advantages of MPMLS are: 1) it can maintain good population diversity so that the search can approach the optimal solution of the original problem from different directions and 2) it only needs to build local surrogates so that the computational overhead of the model building can be reduced. Numerical experiments demonstrate that our proposed algorithm performs much better than some other state-of-the-art evolutionary algorithms.

A two-phase evolutionary algorithm is developed to find multiple solutions of a nonlinear equations system. It transforms a nonlinear equations system into a multimodal optimization problem. In phase one of the proposed algorithm, a strategy combines a multiobjective optimization technique and a niching technique to maintain the population diversity. Phase two consists of a detection method and a local search method for encouraging the convergence. The detection method finds several promising subregions and the local search method locates the corresponding optimal solutions in each promising subregion. The experiments on a set of 30 nonlinear equation systems demonstrate that the proposed algorithm is better than other state-of-the-art algorithms.

The surrogate-assisted optimization algorithms (SAOAs) are very promising for solving computationally expensive optimization problems (EOPs). Generally, the performance of a SAOA is determined by the quality of its surrogate model and the infill sampling criterion. In this paper, we propose a radial basis function (RBF) assisted optimization algorithm with batch infill sampling criterion for solving EOPs (short for RBFBS). In RBFBS, the quality of RBF model is adjusted by choosing a good shape parameter via solving a sub-expensive hyperparameter optimization problem. Moreover, a batch infill sampling criterion that includes a bi-objective-based sampling approach and a single-objective-based sampling approach is proposed to get a batch of samples for expensive evaluation. The experimental results on various benchmark problems show that RBFBS is very promising for expensive optimization.