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Pointwise expected hypervolume improvement for expensive multi-objective optimization

JOURNAL OF GLOBAL optimization 2025年第1期91卷 171-197页

作者： Mei, Li Zhan, Dawei Southwest Jiaotong Univ Sch Comp & Artificial Intelligence Chengdu 611756 Sichuan Peoples R China

The expected hypervolume improvement (EHVI) is a popular infill criterion used in the multi-objective efficient global optimization algorithm for solving expensive multi-objective optimization problems. However, its exact calculation is complex and time-consuming when the number of objectives is larger than three, which prohibits its usage in problems with more than three objectives. To tackle this problem, we propose a new simple and fast criterion called pointwise expected hypervolume improvement (pEHVI) in this work. In the proposed criterion, we first compute the EHVI values of the studying point by considering one non-dominated front point at a time, and then take the minimum of these EHVI values as the final measurement. The proposed pEHVI is derived in closed-form expression and has linear time complexity with respect to the number of objectives. In addition, we theoretically prove the monotonicity and convergence properties of the proposed pEHVI criterion in this work. Compared with the traditional expected hypervolume improvement, the new infill criterion is significantly faster to compute, especially when the number of objectives is larger than three. Numerical experiments show that the proposed criterion is also able to achieve competitive optimization performance compared with five state-of-the-art multi-objective efficient global optimization algorithms. This work provides a fast and efficient hypervolume-based expected improvement infill criterion for expensive multi-objective optimization.

关键词： Expected hypervolume improvement expensive multi-objective optimization Kriging model Efficient global optimization

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Heterogeneous approximation-assisted search for expensive multi-objective optimization

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SWARM AND EVOLUTIONARY COMPUTATION 2025年 95卷

作者： Qin, Shufen Sun, Chaoli Taiyuan Univ Sci & Technol Sch Comp Sci & Technol Taiyuan 030024 Peoples R China

The cheap surrogate model is commonly used to guide the multi-objective optimization algorithm in the search for the optimum of the expensive optimization problem. However, modeling diversity and its quality are the keys that affect the performance of approximating the original problem. Using multiple heterogeneous models can provide more diverse approximations for complicated optimization problems. Meanwhile, the location relationship between individuals and training samples is a potential benefit for selecting infill individuals to update the model. Therefore, this paper proposes to train two heterogeneous models for each expensive objection function, with the update of the models using the promising individuals based on the approximated domination relationship and the crowding distance between individuals and evaluated samples. Differently, the function estimation of each individual is the sum of two predicted values in a probability-weighted way together with its uncertainty. In addition, the promising individuals are selected by the dominant numbers or the distance to the decision domain center and the crowding distance to the neighbors, otherwise adopting the difference in convergence and crowding distance between all candidates and the training samples to select the individual for expensive function evaluations if the training set dominates all offspring individuals. Experimental studies analyze the effectiveness of the heterogeneous approximation-based guiding search and examine the superiority of the proposed algorithm compared to five recent epidemic optimization algorithms for DTLZ, WFG benchmark problems, and a practical application.

关键词： expensive multi-objective optimization Surrogate-assisted evolutionary optimization Heterogeneous approximation Dominance and distance-based infill sampling

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SEAMS: A surrogate-assisted evolutionary algorithm with metric-based dynamic strategy for expensive multi-objective optimization

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EXPERT SYSTEMS WITH APPLICATIONS 2025年 265卷

作者： Liu, Haitao Wang, Chia-Hung Fujian Univ Technol Coll Comp Sci & Math 69 Xuefu South Rd Fuzhou City 350118 Fujian Province Peoples R China Fujian Prov Key Lab Big Data Min & Applicat Fuzhou 350118 Fujian Peoples R China

In real-world scenarios where resources for evaluating expensive optimization problems are limited and the reliability of trained models is hard to assess, the quality of the non-dominated front formed by algorithms tends to below. This paper proposes a metric-based surrogate-assisted evolutionary algorithm for multi-objective expensive optimization, incorporating a novel model management strategy that integrates a regeneration mechanism. This approach aims to achieve a well-balanced convergence and diversity, facilitating the attainment of high-quality non-dominated fronts to address expensive multi-objective optimization problems. The model management strategy, based on metrics, comprehensively evaluates the reliability of the classification model and selects appropriate strategies for offspring selection. Moreover, through significance analysis of the population, the regeneration mechanism identifies high-quality dimensions for regenerating offspring. The algorithm maximizes the utilization of the classification model to guide the generation and selection of offspring in the population. Experiments on DTLZ, MaF, WFG, and the high-dimensional portfolio optimization problem demonstrate that the proposed algorithm outperforms nine state-of-the-art surrogate-assisted evolutionary algorithms, highlighting its superior performance across various scenarios.

关键词： Evolutionary algorithms Surrogate-assisted optimization expensive multi-objective optimization Classification Model management Front modeling

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Bi-Level Model Management Strategy for Solving expensive multi-objective optimization Problems

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IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTATIONAL INTELLIGENCE 2025年第1期9卷 332-346页

作者： Li, Fei Yang, Yujie Liu, Yuhao Liu, Yuanchao Qian, Muyun Anhui Univ Technol Dept Elect & Informat Engn Anhui Prov Key Lab Met Engn & Resources Recycling Anhui Prov Engn Lab Intelligent D/emolit Equipment Maanshan 243002 Anhui Peoples R China Anhui Univ Technol Sch Elect & Informat Engn Maanshan 243032 Peoples R China Northeastern Univ State Key Lab Synthet Automat Proc Ind Shenyang 110819 Peoples R China Anhui Univ Technol Dept Elect & Informat Engn Maanshan 243002 Anhui Peoples R China

Model management strategy is the main component of surrogate-assisted evolutionary algorithms for solving expensive multi-objective optimization problems(EMOPs). In such problems, evaluating the true fitness function requires significant computational resources, which necessitates effective determination of which individuals should be selected for evaluation. However, existing model management strategies often struggle to effectively balance exploration and exploitation when selecting individuals. To mitigate this issue, a bi-level model management strategy is proposed. The selection procedure not only considers exploring the objective space by balancing predicted values and uncertainty in the lower-level but also considers convergence and diversity in the upper-level selection. Specifically, we first divide the objective space into some sub-spaces through a set of direction vectors. Then, we select the first rank individuals via adopting the non-dominated sorting to balance the predicted objective values and the uncertainty in each subspace. The intersection individuals of the selected candidates can be denoted as the lower-level solutions. In the upper-level selection, we combine the modified inverted generational distance (IGD(+)) and shift-based density estimation (SDE) indicators to select the most promising individuals from the lower-level to update the model. Experimental results on benchmark instances show that the proposed algorithm is competitive compared with some representative algorithms.

关键词： optimization Uncertainty Evolutionary computation Vectors Predictive models Convergence Load modeling Surrogate-assisted evolutionary algorithm expensive multi-objective optimization bi-level model management strategy performance indicators Kriging models

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A New multitask Joint Learning Framework for expensive multi-objective optimization Problems

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IEEE TRANSACTIONS ON EMERGING TOPICS IN COMPUTATIONAL INTELLIGENCE 2024年第2期8卷 1894-1909页

作者： Luo, Jianping Dong, Yongfei Liu, Qiqi Zhu, Zexuan Cao, Wenming Tan, Kay Chen Jin, Yaochu Shenzhen Univ Guangdong Key Lab Intelligent Informat Proc Shenzhen 518060 Peoples R China Shenzhen Univ Coll Elect & Informat Engn Shenzhen Key Lab Media Secur Shenzhen 518060 Peoples R China Westlake Univ Sch Engn Hangzhou 310030 Peoples R China Shenzhen Univ Coll Comp Sci & Software Engn Shenzhen 518060 Peoples R China Hong Kong Polytech Univ Dept Comp Hong Kong Peoples R China

In this paper, we propose a multi-objective optimization algorithm based on multitask conditional neural processes (MTCNPs) to deal with expensive multi-objective optimization problems (MOPs). In the proposed algorithm, an MOP is decomposed into several subproblems. Several related subproblems are assigned to a task group and jointly handled using an MTCNPs surrogate model, in which multi-task learning is incorporated to exploit the similarity across the subproblems via joint surrogate model learning. Each subproblem in a task group is modeled by a conditional neural processes (CNPs) instead of a Gaussian Process (GP), thus avoiding the calculation of the GP covariance matrix. In addition, multiple subproblems are jointly learned through a multi-layer similarity network with activation function, which can measure and utilize the similarity and useful information among subproblems more effectively and improve the accuracy and robustness of the surrogate model. Experimental studies under several scenarios indicate that the proposed algorithm performs better than several state-of-the-art multi-objective evolutionary algorithms for expensive MOPs. The parameter sensitivity and effectiveness of the proposed algorithm are analyzed in detail.

关键词： optimization Task analysis Correlation Evolutionary computation Data models Covariance matrices Computational modeling multitask processes conditional neural processes neural networks expensive multi-objective optimization

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A clustering-based surrogate-assisted evolutionary algorithm (CSMOEA) for expensive multi-objective optimization

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SOFT COMPUTING 2023年第15期27卷 10665-10686页

作者： Wang, Wenxin Dong, Huachao Wang, Peng Wang, Xinjing Shen, Jiangtao Northwestern Polytech Univ Sch Marine Sci & Technol Xian 710068 Peoples R China

This paper presents a novel surrogate-assisted evolutionary algorithm, CSMOEA, for multi-objective optimization problems (MOPs) with computationally expensive objectives. Considering most surrogate-assisted evolutionary algorithms (SAEAs) do not make full use of population information and only use population information in either the objective space or the design space independently, to address this limitation, we propose a new strategy for comprehensive utilization of population information of objective and design space. The proposed CSMOEA adopts an adaptive clustering strategy to divide the current population into good and bad groups, and the clustering centers in the design space are obtained, respectively. Then, a bi-level sampling strategy is proposed to select the best samples in both the design and objective space, using distance to the clustering centers and approximated objective values of radial basis functions. The effectiveness of CSMOEA is compared with five state-of-the-art algorithms on 21 widely used benchmark problems, and the results show high efficiency and a good balance between convergence and diversity. Additionally, CSMOEA is applied to the shape optimization of blend-wing-body underwater gliders with 14 decision variables and two objectives, demonstrating its effectiveness in solving real-world engineering problems.

关键词： expensive multi-objective optimization Radial basis function Bi-level sampling strategy Clustering algorithm

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A bagging-based surrogate-assisted evolutionary algorithm for expensive multi-objective optimization

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NEURAL COMPUTING & APPLICATIONS 2022年第14期34卷 12097-12118页

作者： Liu, Yuanchao Liu, Jianchang Tan, Shubin Yang, Yongkuan Li, Fei Northeastern Univ State Key Lab Synthet Automat Proc Ind Shenyang Peoples R China Northeastern Univ Coll Informat Sci & Engn Shenyang Peoples R China Xiamen Univ Technol Sch Elect Engn & Automat Xiamen Peoples R China Anhui Univ Technol Dept Elect & Informat Engn Maanshan Anhui Peoples R China

It is a big challenge for multi-objective evolutionary algorithms to solve expensive multi-objective optimization due to high computational cost. To effectively address expensive multi-objective optimization, this work proposes a novel surrogate-assisted evolutionary algorithm (SAEA), named bagging-based SAEA (B-SAEA). In the proposed method, bagging is introduced to construct high-quality surrogate ensembles for each expensive objective under a limited number of training points. Thereafter, an evolutionary search is applied to fully search for the constructed surrogate ensembles with the help of generation-based search strategy. Thus, surrogate ensembles and evolutionary search can be seamlessly integrated. In addition, a niche-based infill solutions selection strategy is proposed to select the promising points as the infill solutions for real fitness evaluations. As a result, a good balance between convergence and diversity can be achieved within a limited computational budget. Experimental results on commonly used benchmark test problems and real-world engineering application have demonstrated that the proposed method performs competitively compared with other state-of-the-art methods.

关键词： expensive multi-objective optimization Surrogate-assisted evolutionary algorithm Bagging Niche-based infill solutions selection strategy

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RM-SAEA: Regularity Model Based Surrogate-Assisted Evolutionary Algorithms for expensive multi-objective optimization 23

RM-SAEA: Regularity Model Based Surrogate-Assisted Evolution...

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Lu, Yongfan Li, Bingdong Qian, Hong Hong, Wenjing Yang, Peng Zhou, Aimin East China Normal Univ Shanghai Inst AI Educ Shanghai Peoples R China East China Normal Univ Sch Comp Sci & Technol Shanghai Peoples R China Southern Univ Sci & Technol Dept Comp Sci & Engn Guangdong Prov Key Lab Brain Inspired Intelligent Shenzhen Peoples R China

ISBN: (纸本)9798400701191

Due to computationally and/or financially costly evaluation, tackling expensive multi-objective optimization problems is quite challenging for evolutionary algorithms. One popular approach to these problems is building cheap surrogate models to replace the expensive real function evaluations. To this end, various kinds of surrogate-assisted evolutionary algorithms (SAEAs) have been proposed, building surrogate models which predict the fitness values, classifications, or relation of the candidate solutions. However, offspring generation, despite its important role in evolutionary optimization, has not received enough attention in these SAEAs. In this paper, a regularity model based framework, namely RM-SAEA, is proposed for better offspring generation in expensive multi-objective optimization. To be specific, RM-SAEA is featured with a heterogeneous offspring generation module, which is composed of a regularity model and a general genetic operator. Moreover, in order to alleviate the data deficiency issue in the expensive optimization scenario, a data augmentation strategy is employed while training the regularity model. Finally, two representative SAEAs are embedded into RM-SAEA in order to instantiate the proposed framework. Experimental results on benchmark multi-objective problems with up to 10 objectives demonstrate that RM-SAEA achieves the best overall performance compared with 6 state-of-the-art algorithms.

关键词： expensive multi-objective optimization regularity model Pareto set learning surrogate-assisted evolutionary algorithm

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Batch subproblem coevolution with gaussian process-driven linear models for expensive multi-objective optimization

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SWARM AND EVOLUTIONARY COMPUTATION 2024年 91卷

作者： Wang, Zhenkun Chen, Yuanyao Li, Genghui Xie, Lindong Zhang, Yu Southern Univ Sci & Technol Sch Syst Design & Intelligent Mfg Shenzhen Guangdong Peoples R China Southern Univ Sci & Technol Dept Comp Sci & Engn Shenzhen Guangdong Peoples R China Shenzhen Univ Coll Comp Sci & Software Engn Shenzhen Guangdong Peoples R China

The efficacy of surrogate-assisted multi-objective evolutionary algorithms (SAMOEAs) in addressing expensive multi-objective optimization problems (MOPs) is contingent upon the modeling techniques and model-based infill sampling strategies. In addressing this pivotal aspect, this paper introduces a pioneering methodology known as batch subproblem coevolution with Gaussian process-driven linear models (BSCo-GPLM). Specifically, from a modeling perspective, BSCo-GPLM decomposes the MOP into single-objective subproblems. Following this decomposition, for each subproblem, a Gaussian process-driven linear model (GPLM) is collaboratively trained to prevent overfitting and improve prediction accuracy. Regarding infill sampling, collaborative optimization of all GPLMs yields optimal candidate solutions for each subproblem, organized into coherent clusters. Within each cluster, only the solution with the highest utility is evaluated. Relying on the heightened prediction accuracy of the GPLM model and an efficient batch sampling strategy, BSCo-GPLM exhibits clear superiority over state-of-the-art SAMOEAs in effectively addressing expensive MOPs. The source code of BSCo-GPLM is available at https://***/CIAM-Group/EvolutionaryAlgorithm_Codes/tree/main/BSCoGPLM.

关键词： Surrogate-assisted evolutionary optimization Gaussian process-driven linear model Batch subproblem coevolution expensive multi-objective optimization

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A trust-region-like algorithm for expensive multi-objective optimization

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APPLIED SOFT COMPUTING 2023年 148卷

作者： Liu, Hongwei Zhou, Changcong Liu, Fuchao Duan, Zunyi Zhao, Haodong Northwestern Polytech Univ Dept Engn Mech Xian 710072 Peoples R China Harbin Inst Technol Natl Key Lab Sci & Technol Adv Composites Special Harbin 150080 Peoples R China Northwestern Polytech Univ Shenzhen Res & Dev Inst Shenzhen 518063 Peoples R China

In solving expensive multi-objective optimization problems, surrogate models have been widely investigated. The existing multi-objective algorithms adopting surrogate models can be classified into two categories: surrogate-assisted evolutionary algorithms (SAEAs) and surrogate-based optimization algorithms (SBAs). How-ever, their efficiency and convergence remain considerably inadequate. In this work, we propose a trust -region-like algorithm for dealing with expensive multi-objective problems, where only a small number of expensive objective function evaluations are allowed. The optimization process of the proposed algorithm mainly includes two stages. The first stage is the surrogate-assisted stage, in which a promising population with the consideration of convergence and diversity is obtained by reference vector-guided evolutionary algorithms (RVEA). The promising population is considered the preliminary trust-region. Moreover, a new adaptive sampling selection criterion switching between two different sampling strategies is used to further narrow the trust-region. The second stage is the surrogate-based stage, wherein the selection of the most promising individuals is carried out from the ultimate trust-region using the pseudo hypervolume-based expected improvement matrix (PEIM) criterion. Comparison results on the benchmark functions demonstrate that the proposed algorithm is competitive with five state-of-art algorithms in a limited computational budget. Finally, the proposed algorithm is used in the design optimization of a variable stiffness composite cylinder.

关键词： expensive multi-objective optimization Surrogate models Sampling strategies Trust -region Reference vectors

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