Hisao Ishibuchi

Naoki Masuyama, Yusuke Nojima, Yuichiro Toda et al.

With the increasing importance of data privacy protection, various privacy-preserving machine learning methods have been proposed. In the clustering domain, various algorithms with a federated learning framework (i.e., federated clustering) have been actively studied and showed high clustering performance while preserving data privacy. However, most of the base clusterers (i.e., clustering algorithms) used in existing federated clustering algorithms need to specify the number of clusters in advance. These algorithms, therefore, are unable to deal with data whose distributions are unknown or continually changing. To tackle this problem, this paper proposes a privacy-preserving continual federated clustering algorithm. In the proposed algorithm, an adaptive resonance theory-based clustering algorithm capable of continual learning is used as a base clusterer. Therefore, the proposed algorithm inherits the ability of continual learning. Experimental results with synthetic and real-world datasets show that the proposed algorithm has superior clustering performance to state-of-the-art federated clustering algorithms while realizing data privacy protection and continual learning ability. The source code is available at \url{https://github.com/Masuyama-lab/FCAC}.

Rongguang Ye, Longcan Chen, Wei-Bin Kou et al.

Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high performance of the PSL methods. Designing a sampling strategy of preference vectors is difficult since the Pareto front shape cannot be known in advance. To make Pareto set learning work effectively in any Pareto front shape, we propose a Pareto front shape-agnostic Pareto Set Learning (GPSL) that does not require the prior information about the Pareto front. The fundamental concept behind GPSL is to treat the learning of the Pareto set as a distribution transformation problem. Specifically, GPSL can transform an arbitrary distribution into the Pareto set distribution. We demonstrate that training a neural network by maximizing hypervolume enables the process of distribution transformation. Our proposed method can handle any shape of the Pareto front and learn the Pareto set without requiring prior knowledge. Experimental results show the high performance of our proposed method on diverse test problems compared with recent Pareto set learning algorithms.

Naoki Masuyama, Yusuke Nojima, Stefan Wermter et al.

Online clustering methods that adaptively create and update nodes as data arrive often make node learning explicit, whereas the mapping from the learned node state to output clusters often remains implicit or simplified. Implicit mappings make output clusters sensitive to weak graph bridges or local relations based on distance in the graph over learned nodes, leaving no explicit constraint on which node groups remain intact during mapping. This paper addresses this gap by proposing PHIDA, a persistence-guided node-to-cluster mapping method for online clustering with learned nodes. PHIDA implements this mapping within Adaptive Resonance Theory (ART)-based online clustering by combining Inverse-Distance ART (IDA) node learning with node-to-cluster mapping constrained by Persistent Homology (PH). Experiments on 24 benchmark datasets show that PHIDA achieves the best average ranks in stationary comparisons that include the recent stationary-only clustering methods, while also improving aggregate performance in the nonstationary setting over the evaluated online methods that adaptively create and update nodes. Ablations and comparisons with conventional node-to-cluster mappings indicate that the observed gains are associated with PH-constrained mapping that preserves raw PH components, together with the use of the PH component view during node learning. Source code is available at https://github.com/Masuyama-lab/PHIDA

Rongguang Ye, Longcan Chen, Jinyuan Zhang et al.

Recently, Pareto Set Learning (PSL) has been proposed for learning the entire Pareto set using a neural network. PSL employs preference vectors to scalarize multiple objectives, facilitating the learning of mappings from preference vectors to specific Pareto optimal solutions. Previous PSL methods have shown their effectiveness in solving artificial multi-objective optimization problems (MOPs) with uniform preference vector sampling. The quality of the learned Pareto set is influenced by the sampling strategy of the preference vector, and the sampling of the preference vector needs to be decided based on the Pareto front shape. However, a fixed preference sampling strategy cannot simultaneously adapt the Pareto front of multiple MOPs. To address this limitation, this paper proposes an Evolutionary Preference Sampling (EPS) strategy to efficiently sample preference vectors. Inspired by evolutionary algorithms, we consider preference sampling as an evolutionary process to generate preference vectors for neural network training. We integrate the EPS strategy into five advanced PSL methods. Extensive experiments demonstrate that our proposed method has a faster convergence speed than baseline algorithms on 7 testing problems. Our implementation is available at https://github.com/rG223/EPS.

Hiroki Shiraishi, Yohei Hayamizu, Tomonori Hashiyama et al.

Rule representations significantly influence the search capabilities and decision boundaries within the search space of Learning Classifier Systems (LCSs), a family of rule-based machine learning systems that evolve interpretable models through evolutionary processes. However, it is very difficult to choose an appropriate rule representation for each problem. Additionally, some problems benefit from using different representations for different subspaces within the input space. Thus, an adaptive mechanism is needed to choose an appropriate rule representation for each rule in LCSs. This article introduces a flexible rule representation using a four-parameter beta distribution and integrates it into a fuzzy-style LCS. The four-parameter beta distribution can form various function shapes, and this flexibility enables our LCS to automatically select appropriate representations for different subspaces. Our rule representation can represent crisp/fuzzy decision boundaries in various boundary shapes, such as rectangles and bells, by controlling four parameters, compared to the standard representations such as trapezoidal ones. Leveraging this flexibility, our LCS is designed to adapt the appropriate rule representation for each subspace. Moreover, our LCS incorporates a generalization bias favoring crisp rules where feasible, enhancing model interpretability without compromising accuracy. Experimental results on real-world classification tasks show that our LCS achieves significantly superior test accuracy and produces more compact rule sets. Our implementation is available at https://github.com/YNU-NakataLab/Beta4-UCS. An extended abstract related to this work is available at https://doi.org/10.36227/techrxiv.174900805.59801248/v1.

Hiroki Shiraishi, Hisao Ishibuchi, Masaya Nakata

The decision-making process significantly influences the predictions of machine learning models. This is especially important in rule-based systems such as Learning Fuzzy-Classifier Systems (LFCSs) where the selection and application of rules directly determine prediction accuracy and reliability. LFCSs combine evolutionary algorithms with supervised learning to optimize fuzzy classification rules, offering enhanced interpretability and robustness. Despite these advantages, research on improving decision-making mechanisms (i.e., class inference schemes) in LFCSs remains limited. Most LFCSs use voting-based or single-winner-based inference schemes. These schemes rely on classification performance on training data and may not perform well on unseen data, risking overfitting. To address these limitations, this article introduces a novel class inference scheme for LFCSs based on the Dempster-Shafer Theory of Evidence (DS theory). The proposed scheme handles uncertainty well. By using the DS theory, the scheme calculates belief masses (i.e., measures of belief) for each specific class and the ``I don't know'' state from each fuzzy rule and infers a class from these belief masses. Unlike the conventional schemes, the proposed scheme also considers the ``I don't know'' state that reflects uncertainty, thereby improving the transparency and reliability of LFCSs. Applied to a variant of LFCS (i.e., Fuzzy-UCS), the proposed scheme demonstrates statistically significant improvements in terms of test macro F1 scores across 30 real-world datasets compared to conventional voting-based and single-winner-based fuzzy inference schemes. It forms smoother decision boundaries, provides reliable confidence measures, and enhances the robustness and generalizability of LFCSs in real-world applications. Our implementation is available at https://github.com/YNU-NakataLab/jUCS.

Naoki Masuyama, Yuichiro Toda, Yusuke Nojima et al.

Clustering in stationary and nonstationary settings, where data distributions remain static or evolve over time, requires models that can adapt to distributional shifts while preserving previously learned cluster structures. This paper proposes an Adaptive Resonance Theory (ART)-based topological clustering algorithm that autonomously adjusts its recalculation interval and vigilance threshold through a diversity-driven adaptation mechanism. This mechanism enables hyperparameter-free learning that maintains cluster stability and continuity in dynamic environments. Experiments on 24 real-world datasets demonstrate that the proposed algorithm outperforms state-of-the-art methods in both clustering performance and continual learning capability. These results highlight the effectiveness of the proposed parameter adaptation in mitigating catastrophic forgetting and maintaining consistent clustering in evolving data streams. Source code is available at https://github.com/Masuyama-lab/IDAT

Hiroki Shiraishi, Hisao Ishibuchi, Masaya Nakata

Function approximation is a critical task in various fields. However, existing neural network approaches struggle with locally complex or discontinuous functions due to their reliance on a single global model covering the entire problem space. We propose X-KAN, a novel method that optimizes multiple local Kolmogorov-Arnold Networks (KANs) through an evolutionary rule-based machine learning framework called XCSF. X-KAN combines KAN's high expressiveness with XCSF's adaptive partitioning capability by implementing local KAN models as rule consequents and defining local regions via rule antecedents. Our experimental results on artificial test functions and real-world datasets demonstrate that X-KAN significantly outperforms conventional methods, including XCSF, Multi-Layer Perceptron, and KAN, in terms of approximation accuracy. Notably, X-KAN effectively handles functions with locally complex or discontinuous structures that are challenging for conventional KAN, using a compact set of rules (average 7.2 $\pm$ 2.3 rules). These results validate the effectiveness of using KAN as a local model in XCSF, which evaluates the rule fitness based on both accuracy and generality. Our X-KAN implementation is available at https://github.com/YNU-NakataLab/X-KAN.

Naoki Masuyama, Takanori Takebayashi, Yusuke Nojima et al.

In general, a similarity threshold (i.e., a vigilance parameter) for a node learning process in Adaptive Resonance Theory (ART)-based algorithms has a significant impact on clustering performance. In addition, an edge deletion threshold in a topological clustering algorithm plays an important role in adaptively generating well-separated clusters during a self-organizing process. In this paper, we propose an ART-based topological clustering algorithm that integrates parameter estimation methods for both the similarity threshold and the edge deletion threshold. The similarity threshold is estimated using a determinantal point process-based criterion, while the edge deletion threshold is defined based on the age of edges. Experimental results with synthetic and real-world datasets show that the proposed algorithm has superior clustering performance to state-of-the-art clustering algorithms without requiring parameter specifications specific to the datasets. Source code is available at https://github.com/Masuyama-lab/CAE

Rongguang Ye, Lei Chen, Weiduo Liao et al.

Pareto front learning is a technique that introduces preference vectors in a neural network to approximate the Pareto front. Previous Pareto front learning methods have demonstrated high performance in approximating simple Pareto fronts. These methods often sample preference vectors from a fixed Dirichlet distribution. However, no fixed sampling distribution can be adapted to diverse Pareto fronts. Efficiently sampling preference vectors and accurately estimating the Pareto front is a challenge. To address this challenge, we propose a data-driven preference vector sampling framework for Pareto front learning. We utilize the posterior information of the objective functions to adjust the parameters of the sampling distribution flexibly. In this manner, the proposed method can sample preference vectors from the location of the Pareto front with a high probability. Moreover, we design the distribution of the preference vector as a mixture of Dirichlet distributions to improve the performance of the model in disconnected Pareto fronts. Extensive experiments validate the superiority of the proposed method compared with state-of-the-art algorithms.

Tianye Shu, Ke Shang, Cheng Gong et al.

For a control problem with multiple conflicting objectives, there exists a set of Pareto-optimal policies called the Pareto set instead of a single optimal policy. When a multi-objective control problem is continuous and complex, traditional multi-objective reinforcement learning (MORL) algorithms search for many Pareto-optimal deep policies to approximate the Pareto set, which is quite resource-consuming. In this paper, we propose a simple and resource-efficient MORL algorithm that learns a continuous representation of the Pareto set in a high-dimensional policy parameter space using a single hypernet. The learned hypernet can directly generate various well-trained policy networks for different user preferences. We compare our method with two state-of-the-art MORL algorithms on seven multi-objective continuous robot control problems. Experimental results show that our method achieves the best overall performance with the least training parameters. An interesting observation is that the Pareto set is well approximated by a curved line or surface in a high-dimensional parameter space. This observation will provide insight for researchers to design new MORL algorithms.

Javier Poyatos, Javier Del Ser, Salvador Garcia et al.

In Artificial Intelligence, there is an increasing demand for adaptive models capable of dealing with a diverse spectrum of learning tasks, surpassing the limitations of systems devised to cope with a single task. The recent emergence of General-Purpose Artificial Intelligence Systems (GPAIS) poses model configuration and adaptability challenges at far greater complexity scales than the optimal design of traditional Machine Learning models. Evolutionary Computation (EC) has been a useful tool for both the design and optimization of Machine Learning models, endowing them with the capability to configure and/or adapt themselves to the task under consideration. Therefore, their application to GPAIS is a natural choice. This paper aims to analyze the role of EC in the field of GPAIS, exploring the use of EC for their design or enrichment. We also match GPAIS properties to Machine Learning areas in which EC has had a notable contribution, highlighting recent milestones of EC for GPAIS. Furthermore, we discuss the challenges of harnessing the benefits of EC for GPAIS, presenting different strategies to both design and improve GPAIS with EC, covering tangential areas, identifying research niches, and outlining potential research directions for EC and GPAIS.

Naoki Masuyama, Narito Amako, Yuna Yamada et al.

Adaptive Resonance Theory (ART) is considered as an effective approach for realizing continual learning thanks to its ability to handle the plasticity-stability dilemma. In general, however, the clustering performance of ART-based algorithms strongly depends on the specification of a similarity threshold, i.e., a vigilance parameter, which is data-dependent and specified by hand. This paper proposes an ART-based topological clustering algorithm with a mechanism that automatically estimates a similarity threshold from the distribution of data points. In addition, for improving information extraction performance, a divisive hierarchical clustering algorithm capable of continual learning is proposed by introducing a hierarchical structure to the proposed algorithm. Experimental results demonstrate that the proposed algorithm has high clustering performance comparable with recently-proposed state-of-the-art hierarchical clustering algorithms.

Ke Shang, Tianye Shu, Hisao Ishibuchi

Hypervolume contribution is an important concept in evolutionary multi-objective optimization (EMO). It involves in hypervolume-based EMO algorithms and hypervolume subset selection algorithms. Its main drawback is that it is computationally expensive in high-dimensional spaces, which limits its applicability to many-objective optimization. Recently, an R2 indicator variant (i.e., $R_2^{\text{HVC}}$ indicator) is proposed to approximate the hypervolume contribution. The $R_2^{\text{HVC}}$ indicator uses line segments along a number of direction vectors for hypervolume contribution approximation. It has been shown that different direction vector sets lead to different approximation quality. In this paper, we propose \textit{Learning to Approximate (LtA)}, a direction vector set generation method for the $R_2^{\text{HVC}}$ indicator. The direction vector set is automatically learned from training data. The learned direction vector set can then be used in the $R_2^{\text{HVC}}$ indicator to improve its approximation quality. The usefulness of the proposed LtA method is examined by comparing it with other commonly-used direction vector set generation methods for the $R_2^{\text{HVC}}$ indicator. Experimental results suggest the superiority of LtA over the other methods for generating high quality direction vector sets.

Ke Shang, Tianye Shu, Hisao Ishibuchi et al.

In the evolutionary multi-objective optimization (EMO) field, the standard practice is to present the final population of an EMO algorithm as the output. However, it has been shown that the final population often includes solutions which are dominated by other solutions generated and discarded in previous generations. Recently, a new EMO framework has been proposed to solve this issue by storing all the non-dominated solutions generated during the evolution in an archive and selecting a subset of solutions from the archive as the output. The key component in this framework is the subset selection from the archive which usually stores a large number of candidate solutions. However, most studies on subset selection focus on small candidate solution sets for environmental selection. There is no benchmark test suite for large-scale subset selection. This paper aims to fill this research gap by proposing a benchmark test suite for subset selection from large candidate solution sets, and comparing some representative methods using the proposed test suite. The proposed test suite together with the benchmarking studies provides a baseline for researchers to understand, use, compare, and develop subset selection methods in the EMO field.

Weiyu Chen, Hisao Ishibuchi, Ke Shang

Subset selection is an important component in evolutionary multiobjective optimization (EMO) algorithms. Clustering, as a classic method to group similar data points together, has been used for subset selection in some fields. However, clustering-based methods have not been evaluated in the context of subset selection from solution sets obtained by EMO algorithms. In this paper, we first review some classic clustering algorithms. We also point out that another popular subset selection method, i.e., inverted generational distance (IGD)-based subset selection, can be viewed as clustering. Then, we perform a comprehensive experimental study to evaluate the performance of various clustering algorithms in different scenarios. Experimental results are analyzed in detail, and some suggestions about the use of clustering algorithms for subset selection are derived. Additionally, we demonstrate that decision maker's preference can be introduced to clustering-based subset selection.

Rahul Kumar Sevakula, Nishchal Kumar Verma, Hisao Ishibuchi

Recent discoveries in Deep Neural Networks are allowing researchers to tackle some very complex problems such as image classification and audio classification, with improved theoretical and empirical justifications. This paper presents a novel scheme to incorporate the use of autoencoders in Fuzzy rule classifiers (FRC). Autoencoders when stacked can learn the complex non-linear relationships amongst data, and the proposed framework built towards FRC can allow users to input expert knowledge to the system. This paper further introduces four novel fine-tuning strategies for autoencoders to improve the FRC's classification and rule reduction performance. The proposed framework has been tested across five real-world benchmark datasets. Elaborate comparisons with over 15 previous studies, and across 10-fold cross validation performance, suggest that the proposed methods are capable of building FRCs which can provide state of the art accuracies.

Ke Shang, Hisao Ishibuchi, Weiyu Chen et al.

Hypervolume is widely used in the evolutionary multi-objective optimization (EMO) field to evaluate the quality of a solution set. For a solution set with $μ$ solutions on a Pareto front, a larger hypervolume means a better solution set. Investigating the distribution of the solution set with the largest hypervolume is an important topic in EMO, which is the so-called hypervolume optimal $μ$-distribution. Theoretical results have shown that the $μ$ solutions are uniformly distributed on a linear Pareto front in two dimensions. However, the $μ$ solutions are not always uniformly distributed on a single-line Pareto front in three dimensions. They are only uniform when the single-line Pareto front has one constant objective. In this paper, we further investigate the hypervolume optimal $μ$-distribution in three dimensions. We consider the line- and plane-based Pareto fronts. For the line-based Pareto fronts, we extend the single-line Pareto front to two-line and three-line Pareto fronts, where each line has one constant objective. For the plane-based Pareto fronts, the linear triangular and inverted triangular Pareto fronts are considered. First, we show that the $μ$ solutions are not always uniformly distributed on the line-based Pareto fronts. The uniformity depends on how the lines are combined. Then, we show that a uniform solution set on the plane-based Pareto front is not always optimal for hypervolume maximization. It is locally optimal with respect to a $(μ+1)$ selection scheme. Our results can help researchers in the community to better understand and utilize the hypervolume indicator.

Naoki Masuyama, Yusuke Nojima, Chu Kiong Loo et al.

This paper proposes a multi-label classification algorithm capable of continual learning by applying an Adaptive Resonance Theory (ART)-based clustering algorithm and the Bayesian approach for label probability computation. The ART-based clustering algorithm adaptively and continually generates prototype nodes corresponding to given data, and the generated nodes are used as classifiers. The label probability computation independently counts the number of label appearances for each class and calculates the Bayesian probabilities. Thus, the label probability computation can cope with an increase in the number of labels. Experimental results with synthetic and real-world multi-label datasets show that the proposed algorithm has competitive classification performance to other well-known algorithms while realizing continual learning.

Weiyu Chen, Hisao Ishibuchi, Ke Shang

Subset selection is an interesting and important topic in the field of evolutionary multi-objective optimization (EMO). Especially, in an EMO algorithm with an unbounded external archive, subset selection is an essential post-processing procedure to select a pre-specified number of solutions as the final result. In this paper, we discuss the efficiency of greedy subset selection for the hypervolume, IGD and IGD+ indicators. Greedy algorithms usually efficiently handle subset selection. However, when a large number of solutions are given (e.g., subset selection from tens of thousands of solutions in an unbounded external archive), they often become time-consuming. Our idea is to use the submodular property, which is known for the hypervolume indicator, to improve their efficiency. First, we prove that the IGD and IGD+ indicators are also submodular. Next, based on the submodular property, we propose an efficient greedy inclusion algorithm for each indicator. Then, we demonstrate through computational experiments that the proposed algorithms are much faster than the standard greedy subset selection algorithms.

Yiming Peng, Hisao Ishibuchi

Niching is an important and widely used technique in evolutionary multi-objective optimization. Its applications mainly focus on maintaining diversity and avoiding early convergence to local optimum. Recently, a special class of multi-objective optimization problems, namely, multi-modal multi-objective optimization problems (MMOPs), started to receive increasing attention. In MMOPs, a solution in the objective space may have multiple inverse images in the decision space, which are termed as equivalent solutions. Since equivalent solutions are overlapping (i.e., occupying the same position) in the objective space, standard diversity estimators such as crowding distance are likely to select one of them and discard the others, which may cause diversity loss in the decision space. In this study, a general niching mechanism is proposed to make standard diversity estimators more efficient when handling MMOPs. In our experiments, we integrate our proposed niching diversity estimation method into SPEA2 and NSGA-II and evaluate their performance on several MMOPs. Experimental results show that the proposed niching mechanism notably enhances the performance of SPEA2 and NSGA-II on various MMOPs.

Hisao Ishibuchi, Lie Meng Pang, Ke Shang

It is assumed in the evolutionary multi-objective optimization (EMO) community that a final solution is selected by a decision maker from a non-dominated solution set obtained by an EMO algorithm. The number of solutions to be presented to the decision maker can be totally different. In some cases, the decision maker may want to examine only a few representative solutions from which a final solution is selected. In other cases, a large number of non-dominated solutions may be needed to visualize the Pareto front. In this paper, we suggest the use of a general EMO framework with three solution sets to handle various situations with respect to the required number of solutions. The three solution sets are the main population of an EMO algorithm, an external archive to store promising solutions, and a final solution set which is presented to the decision maker. The final solution set is selected from the archive. Thus the population size and the archive size can be arbitrarily specified as long as the archive size is not smaller than the required number of solutions. The final population is not necessarily to be a good solution set since it is not presented to the decision maker. Through computational experiments, we show the advantages of this framework over the standard final population and final archive frameworks. We also discuss how to select a final solution set and how to explain the reason for the selection, which is the first attempt towards an explainable EMO framework.

Ryoji Tanabe, Hisao Ishibuchi

An unbounded external archive (UEA), which stores all nondominated solutions found during the search process, is frequently used to evaluate the performance of multi-objective evolutionary algorithms (MOEAs) in recent studies. A recent benchmarking study also shows that decomposition-based MOEA (MOEA/D) is competitive with state-of-the-art MOEAs when the UEA is incorporated into MOEA/D. However, a parameter study of MOEA/D using the UEA has not yet been performed. Thus, it is unclear how control parameter settings influence the performance of MOEA/D with the UEA. In this paper, we present an analysis of control parameters of MOEA/D under two performance evaluation scenarios. One is a final population scenario where the performance assessment of MOEAs is performed based on all nondominated solutions in the final population, and the other is a reduced UEA scenario where it is based on a pre-specified number of selected nondominated solutions from the UEA. Control parameters of MOEA/D investigated in this paper include the population size, scalarizing functions, and the penalty parameter of the penalty-based boundary intersection (PBI) function. Experimental results indicate that suitable settings of the three control parameters significantly depend on the choice of an optimization scenario. We also analyze the reason why the best parameter setting is totally different for each scenario.

Ryoji Tanabe, Hisao Ishibuchi

A decomposition-based multi-objective evolutionary algorithm with a differential evolution variation operator (MOEA/D-DE) shows high performance on challenging multi-objective problems (MOPs). The DE mutation consists of three key components: a mutation strategy, an index selection method for parent individuals, and a bound-handling method. However, the configuration of the DE mutation operator that should be used for MOEA/D-DE has not been thoroughly investigated in the literature. This configuration choice confuses researchers and users of MOEA/D-DE. To address this issue, we present a review of the existing configurations of the DE mutation operator in MOEA/D-DE and systematically examine the influence of each component on the performance of MOEA/D-DE. Our review reveals that the configuration of the DE mutation operator differs depending on the source code of MOEA/D-DE. In our analysis, a total of 30 configurations (three index selection methods, two mutation strategies, and five bound handling methods) are investigated on 16 MOPs with up to five objectives. Results show that each component significantly affects the performance of MOEA/D-DE. We also present the most suitable configuration of the DE mutation operator, which maximizes the effectiveness of MOEA/D-DE.

Ryoji Tanabe, Hisao Ishibuchi

Multi-modal multi-objective optimization is to locate (almost) equivalent Pareto optimal solutions as many as possible. Some evolutionary algorithms for multi-modal multi-objective optimization have been proposed in the literature. However, there is no efficient method for multi-modal many-objective optimization, where the number of objectives is more than three. To address this issue, this paper proposes a niching indicator-based multi-modal multi- and many-objective optimization algorithm. In the proposed method, the fitness calculation is performed among a child and its closest individuals in the solution space to maintain the diversity. The performance of the proposed method is evaluated on multi-modal multi-objective test problems with up to 15 objectives. Results show that the proposed method can handle a large number of objectives and find a good approximation of multiple equivalent Pareto optimal solutions. The results also show that the proposed method performs significantly better than eight multi-objective evolutionary algorithms.

Ryoji Tanabe, Hisao Ishibuchi

Since around 2000, it has been considered that elitist evolutionary multi-objective optimization algorithms (EMOAs) always outperform non-elitist EMOAs. This paper revisits the performance of non-elitist EMOAs for bi-objective continuous optimization when using an unbounded external archive. This paper examines the performance of EMOAs with two elitist and one non-elitist environmental selections. The performance of EMOAs is evaluated on the bi-objective BBOB problem suite provided by the COCO platform. In contrast to conventional wisdom, results show that non-elitist EMOAs with particular crossover methods perform significantly well on the bi-objective BBOB problems with many decision variables when using the unbounded external archive. This paper also analyzes the properties of the non-elitist selection.

Ryoji Tanabe, Hisao Ishibuchi

Multi-modal multi-objective optimization is to locate (almost) equivalent Pareto optimal solutions as many as possible. While decomposition-based evolutionary algorithms have good performance for multi-objective optimization, they are likely to perform poorly for multi-modal multi-objective optimization due to the lack of mechanisms to maintain the solution space diversity. To address this issue, this paper proposes a framework to improve the performance of decomposition-based evolutionary algorithms for multi-modal multi-objective optimization. Our framework is based on three operations: assignment, deletion, and addition operations. One or more individuals can be assigned to the same subproblem to handle multiple equivalent solutions. In each iteration, a child is assigned to a subproblem based on its objective vector, i.e., its location in the objective space. The child is compared with its neighbors in the solution space assigned to the same subproblem. The performance of improved versions of six decomposition-based evolutionary algorithms by our framework is evaluated on various test problems regarding the number of objectives, decision variables, and equivalent Pareto optimal solution sets. Results show that the improved versions perform clearly better than their original algorithms.

Ryoji Tanabe, Hisao Ishibuchi

Multi-modal multi-objective optimization aims to find all Pareto optimal solutions including overlapping solutions in the objective space. Multi-modal multi-objective optimization has been investigated in the evolutionary computation community since 2005. However, it is difficult to survey existing studies in this field because they have been independently conducted and do not explicitly use the term "multi-modal multi-objective optimization". To address this issue, this paper reviews existing studies of evolutionary multi-modal multi-objective optimization, including studies published under names that are different from "multi-modal multi-objective optimization". Our review also clarifies open issues in this research area.

Ryoji Tanabe, Hisao Ishibuchi

Although synthetic test problems are widely used for the performance assessment of evolutionary multi-objective optimization algorithms, they are likely to include unrealistic properties which may lead to overestimation/underestimation. To address this issue, we present a multi-objective optimization problem suite consisting of 16 bound-constrained real-world problems. The problem suite includes various problems in terms of the number of objectives, the shape of the Pareto front, and the type of design variables. 4 out of the 16 problems are multi-objective mixed-integer optimization problems. We provide Java, C, and Matlab source codes of the 16 problems so that they are available in an off-the-shelf manner. We examine an approximated Pareto front of each test problem. We also analyze the performance of six representative evolutionary multi-objective optimization algorithms on the 16 problems. In addition to the 16 problems, we present 8 constrained multi-objective real-world problems.

Ryoji Tanabe, Hisao Ishibuchi

Although quality indicators play a crucial role in benchmarking evolutionary multi-objective optimization algorithms, their properties are still unclear. One promising approach for understanding quality indicators is the use of the optimal distribution of objective vectors that optimizes each quality indicator. However, it is difficult to obtain the optimal distribution for each quality indicator, especially when its theoretical property is unknown. Thus, optimal distributions for most quality indicators have not been well investigated. To address these issues, first, we propose a problem formulation of finding the optimal distribution for each quality indicator on an arbitrary Pareto front. Then, we approximate the optimal distributions for nine quality indicators using the proposed problem formulation. We analyze the nine quality indicators using their approximated optimal distributions on eight types of Pareto fronts of three-objective problems. Our analysis demonstrates that uniformly-distributed objective vectors over the entire Pareto front are not optimal in many cases. Each quality indicator has its own optimal distribution for each Pareto front. We also examine the consistency among the nine quality indicators.

Lie Meng Pang, Hisao Ishibuchi, Ke Shang

The development of efficient and effective evolutionary multi-objective optimization (EMO) algorithms has been an active research topic in the evolutionary computation community. Over the years, many EMO algorithms have been proposed. The existing EMO algorithms are mainly developed based on the final population framework. In the final population framework, the final population of an EMO algorithm is presented to the decision maker. Thus, it is required that the final population produced by an EMO algorithm is a good solution set. Recently, the use of solution selection framework was suggested for the design of EMO algorithms. This framework has an unbounded external archive to store all the examined solutions. A pre-specified number of solutions are selected from the archive as the final solutions presented to the decision maker. When the solution selection framework is used, EMO algorithms can be designed in a more flexible manner since the final population is not necessarily to be a good solution set. In this paper, we examine the design of MOEA/D under these two frameworks. We use an offline genetic algorithm-based hyper-heuristic method to find the optimal configuration of MOEA/D in each framework. The DTLZ and WFG test suites and their minus versions are used in our experiments. The experimental results suggest the possibility that a more flexible, robust and high-performance MOEA/D algorithm can be obtained when the solution selection framework is used.

Lie Meng Pang, Hisao Ishibuchi, Ke Shang

In the evolutionary multi-objective optimization (EMO) community, it is usually assumed that the final population is presented to the decision maker as the result of the execution of an EMO algorithm. Recently, an unbounded external archive was used to evaluate the performance of EMO algorithms in some studies where a pre-specified number of solutions are selected from all the examined non-dominated solutions. In this framework, which is referred to as the solution selection framework, the final population does not have to be a good solution set. Thus, the solution selection framework offers higher flexibility to the design of EMO algorithms than the final population framework. In this paper, we examine the design of MOEA/D under these two frameworks. First, we show that the performance of MOEA/D is improved by linearly changing the reference point specification during its execution through computational experiments with various combinations of initial and final specifications. Robust and high performance of the solution selection framework is observed. Then, we examine the use of a genetic algorithm-based offline hyper-heuristic method to find the best configuration of MOEA/D in each framework. Finally, we further discuss solution selection after the execution of an EMO algorithm in the solution selection framework.

Hisao Ishibuchi, Lie Meng Pang, Ke Shang

In general, a multi-objective optimization problem does not have a single optimal solution but a set of Pareto optimal solutions, which forms the Pareto front in the objective space. Various evolutionary algorithms have been proposed to approximate the Pareto front using a pre-specified number of solutions. Hundreds of solutions are obtained by their single run. The selection of a single final solution from the obtained solutions is assumed to be done by a human decision maker. However, in many cases, the decision maker does not want to examine hundreds of solutions. Thus, it is needed to select a small subset of the obtained solutions. In this paper, we discuss subset selection from a viewpoint of the final decision making. First we briefly explain existing subset selection studies. Next we formulate an expected loss function for subset selection. We also show that the formulated function is the same as the IGD plus indicator. Then we report experimental results where the proposed approach is compared with other indicator-based subset selection methods.

Yiming Peng, Hisao Ishibuchi

A multi-modal multi-objective optimization problem is a special kind of multi-objective optimization problem with multiple Pareto subsets. In this paper, we propose an efficient multi-modal multi-objective optimization algorithm based on the widely used MOEA/D algorithm. In our proposed algorithm, each weight vector has its own sub-population. With a clearing mechanism and a greedy removal strategy, our proposed algorithm can effectively preserve equivalent Pareto optimal solutions (i.e., different Pareto optimal solutions with same objective values). Experimental results show that our proposed algorithm can effectively preserve the diversity of solutions in the decision space when handling large-scale multi-modal multi-objective optimization problems.

Weiyu Chen, Hisao Ishibuchi, Ke Shang

Recently, the discretization of decision and objective spaces has been discussed in the literature. In some studies, it is shown that the decision space discretization improves the performance of evolutionary multi-objective optimization (EMO) algorithms on continuous multi-objective test problems. In other studies, it is shown that the objective space discretization improves the performance on combinatorial multi-objective problems. However, the effect of the simultaneous discretization of both spaces has not been examined in the literature. In this paper, we examine the effects of the decision space discretization, objective space discretization and simultaneous discretization on the performance of NSGA-II through computational experiments on the DTLZ and WFG problems. Using various settings about the number of decision variables and the number of objectives, our experiments are performed on four types of problems: standard problems, large-scale problems, many-objective problems, and large-scale many-objective problems. We show that the decision space discretization has a positive effect for large-scale problems and the objective space discretization has a positive effect for many-objective problems. We also show the discretization of both spaces is useful for large-scale many-objective problems.

Muhammad Atif, Siddique Latif, Rizwan Ahmad et al.

Cyber-Physical Systems (CPS) allow us to manipulate objects in the physical world by providing a communication bridge between computation and actuation elements. In the current scheme of things, this sought-after control is marred by limitations inherent in the underlying communication network(s) as well as by the uncertainty found in the physical world. These limitations hamper fine-grained control of elements that may be separated by large-scale distances. In this regard, soft computing is an emerging paradigm that can help to overcome the vulnerabilities, and unreliability of CPS by using techniques including fuzzy systems, neural network, evolutionary computation, probabilistic reasoning and rough sets. In this paper, we present a comprehensive contemporary review of soft computing techniques for CPS dependability modeling, analysis, and improvement. This paper provides an overview of CPS applications, explores the foundations of dependability engineering, and highlights the potential role of soft computing techniques for CPS dependability with various case studies, while identifying common pitfalls and future directions. In addition, this paper provides a comprehensive survey on the use of various soft computing techniques for making CPS dependable.

Yuan Yuan, Yew-Soon Ong, Liang Feng et al.

In this report, we suggest nine test problems for multi-task multi-objective optimization (MTMOO), each of which consists of two multiobjective optimization tasks that need to be solved simultaneously. The relationship between tasks varies between different test problems, which would be helpful to have a comprehensive evaluation of the MO-MFO algorithms. It is expected that the proposed test problems will germinate progress the field of the MTMOO research.

Zhaohong Deng, Yizhang Jiang, Fu-Lai Chung et al.

The traditional prototype based clustering methods, such as the well-known fuzzy c-mean (FCM) algorithm, usually need sufficient data to find a good clustering partition. If the available data is limited or scarce, most of the existing prototype based clustering algorithms will no longer be effective. While the data for the current clustering task may be scarce, there is usually some useful knowledge available in the related scenes/domains. In this study, the concept of transfer learning is applied to prototype based fuzzy clustering (PFC). Specifically, the idea of leveraging knowledge from the source domain is exploited to develop a set of transfer prototype based fuzzy clustering (TPFC) algorithms. Three prototype based fuzzy clustering algorithms, namely, FCM, fuzzy k-plane clustering (FKPC) and fuzzy subspace clustering (FSC), have been chosen to incorporate with knowledge leveraging mechanism to develop the corresponding transfer clustering algorithms. Novel objective functions are proposed to integrate the knowledge of source domain with the data of target domain for clustering in the target domain. The proposed algorithms have been validated on different synthetic and real-world datasets and the results demonstrate their effectiveness when compared with both the original prototype based fuzzy clustering algorithms and the related clustering algorithms like multi-task clustering and co-clustering.