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Mathematics
Special Issues
Multi-Objective Optimizations and Their Applications
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Multi-Objective Optimizations and Their Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D2: Operations Research and Fuzzy Decision Making".

Deadline for manuscript submissions: 31 July 2026 | Viewed by 872

Special Issue Editors

Dr. Xilu Wang

E-Mail Website
Guest Editor
School of Computer Science and Electronic Engineering, University of Surrey, Guildford, UK
Interests: multi-objective optimization; data-driven optimization and Bayesian optimization; evolutionary machine learning: neural architecture search, transfer learning
Dr. Jie Tian

E-Mail Website
Guest Editor
School of Information Science and Engineering, Shandong Normal University, Jinan, China
Interests: expensive many-objective optimization; keyhole plasma arc welding; spatial-temporal features; expensive optimization; Bayesian optimization
Dr. Hangyu Zhu

E-Mail Website
Guest Editor
School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, China
Interests: neural architecture search

Special Issue Information

Dear Colleagues,

Multi-objective optimization addresses the fundamental challenge of simultaneously optimizing multiple, often conflicting objectives in engineering design, scientific research, and industrial applications. Unlike single-objective problems, it seeks Pareto-optimal solutions representing optimal trade-offs between competing goals, providing decision-makers with alternatives rather than a single solution. This approach is essential in scenarios where objectives like minimizing cost while maximizing performance cannot be combined without losing critical trade-off information.

Evolutionary multi-objective optimization (EMO) algorithms have emerged as powerful tools due to their population-based nature, naturally aligning with the simultaneous discovery of multiple Pareto-optimal solutions. However, real-world applications present significant computational challenges when objective function evaluations require expensive simulations, physical experiments, or complex models—such as computational fluid dynamics for aerodynamic optimization or finite element analysis for structural design.

Surrogate-assisted and data-driven multi-objective optimization approaches bridge the gap between theoretical capabilities and practical constraints by integrating machine learning models as surrogates for expensive functions and leveraging available historical data, experimental measurements, and computational results. These data-driven methods enable effective design space exploration with dramatically reduced costs while incorporating prior knowledge and uncertainty quantification. Advanced infill strategies, model management, and data fusion techniques ensure judicious surrogate use while maintaining solution quality through the intelligent exploitation of all available information sources. These innovations have transformed multi-objective optimization into a practical tool for complex applications across the aerospace, automotive, drug discovery, and environmental engineering industries.

Dr. Xilu Wang
Dr. Jie Tian
Dr. Hangyu Zhu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • multi-objective optimization
  • surrogate-assisted
  • multi-objective
  • data-driven optimization

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Published Papers (2 papers)

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Research

22 pages, 649 KB  
Article
CoEGAN-BO: Synergistic Co-Evolution of GANs and Bayesian Optimization for High-Dimensional Expensive Many-Objective Problems
by Jie Tian, Hongli Bian, Yuyao Zhang, Xiaoxu Zhang and Hui Liu
Mathematics 2025, 13(21), 3444; https://doi.org/10.3390/math13213444 - 29 Oct 2025
Viewed by 21
Abstract
Bayesian optimization (BO) struggles with data scarcity and poor scalability in high-dimensional many-objective optimization problems. To address this, we propose Co-Evolutionary GAN–Bayesian Optimization (CoEGAN-BO), a novel framework that synergizes generative adversarial networks (GANs) with Bayesian co-evolutionary search for data-driven optimization. The GAN module [...] Read more.
Bayesian optimization (BO) struggles with data scarcity and poor scalability in high-dimensional many-objective optimization problems. To address this, we propose Co-Evolutionary GAN–Bayesian Optimization (CoEGAN-BO), a novel framework that synergizes generative adversarial networks (GANs) with Bayesian co-evolutionary search for data-driven optimization. The GAN module generates synthetic samples conditioned on promising regions identified by BO, while a co-evolutionary mechanism maintains two interacting populations: one explores the GAN’s latent space for diversity, and the other exploits BO’s probabilistic model for convergence. A bi-stage infilling strategy further enhances efficiency: early iterations prioritize exploration via Lp-norm-based candidate selection, later switching to a max–min distance criterion for Pareto refinement. Experiments on expensive multi/many-objective benchmarks show that CoEGAN-BO outperforms four state-of-the-art surrogate-assisted algorithms, achieving superior convergence and diversity under limited evaluation budgets. Full article
(This article belongs to the Special Issue Multi-Objective Optimizations and Their Applications)
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13 pages, 473 KB  
Article
Multi-Objective Batch Energy-Entropy Acquisition Function for Bayesian Optimization
by Hangyu Zhu and Xilu Wang
Mathematics 2025, 13(17), 2894; https://doi.org/10.3390/math13172894 - 8 Sep 2025
Viewed by 676
Abstract
Bayesian Optimization (BO) provides an efficient framework for optimizing expensive black-box functions by employing a surrogate model (typically a Gaussian Process) to approximate the objective function and an acquisition function to guide the search for optimal points. Batch BO extends this paradigm by [...] Read more.
Bayesian Optimization (BO) provides an efficient framework for optimizing expensive black-box functions by employing a surrogate model (typically a Gaussian Process) to approximate the objective function and an acquisition function to guide the search for optimal points. Batch BO extends this paradigm by selecting and evaluating multiple candidate points simultaneously, which improves computational efficiency but introduces challenges in optimizing the resulting high-dimensional acquisition functions. Among existing acquisition functions for batch Bayesian Optimization, entropy-based methods are considered to be state-of-the-art methods due to their ability to enable more globally efficient while avoiding redundant evaluations. However, they often fail to fully capture the dependencies and interactions among the selected batch points. In this work, we propose a Multi-Objective Batch Energy–Entropy acquisition function for Bayesian Optimization (MOBEEBO), which adaptively exploits the correlations among batch points. In addition, MOBEEBO incorporates multiple types of acquisition functions as objectives in a unified framework to achieve more effective batch diversity and quality. Empirical results demonstrate that the proposed algorithm is applicable to a wide range of optimization problems and achieves competitive performance. Full article
(This article belongs to the Special Issue Multi-Objective Optimizations and Their Applications)
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