Evolutionary Multimodal Optimization Research Articles

Peak Identification in Evolutionary Multimodal Optimization: Model, Algorithms, and Metrics.

In this paper, we present a two-phase multimodal optimization model designed to efficiently and accurately identify multiple optima. The first phase employs a population-based search algorithm to locate potential optima, while the second phase introduces a novel peak identification (PI) procedure to filter out non-optimal solutions, ensuring that each identified solution represents a distinct optimum. This approach not only enhances the effectiveness of multimodal optimization but also addresses the issue of redundant solutions prevalent in existing algorithms. We propose two PI algorithms: HVPI, which uses a hill-valley approach to distinguish between optima, without requiring prior knowledge of niche radii; and HVPIC, which integrates HVPI with bisecting K-means clustering to reduce the number of fitness evaluations (FEs). The performance of these algorithms was evaluated using the F-measure, a comprehensive metric that accounts for both the accuracy and redundancy in the solution set. Extensive experiments on a suite of benchmark functions and engineering problems demonstrated that our proposed algorithms achieved a high precision and recall, significantly outperforming traditional methods.

Evolutionary Multimodal Multiobjective Optimization for Traveling Salesman Problems

Multimodal multiobjective optimization problems (MMOPs) are commonly seen in real-world applications. Many evolutionary algorithms have been proposed to solve continuous MMOPs. However, little effort has been made to solve combinatorial (or discrete) MMOPs. Searching for equivalent Pareto optimal solutions in the discrete decision space is challenging. Moreover, the true Pareto optimal solutions of a combinatorial MMOP are usually difficult to know, which has limited the development of its optimizer. In this paper, we first propose a test problem generator for multimodal multiobjective traveling salesman problems (MMTSPs). It can readily generate MMTSPs with known Pareto optimal solutions. Then we propose a novel evolutionary algorithm to solve MMTSPs. In our proposed algorithm, we develop two new edge assembly crossover operators, which are specialized in searching for superior solutions to MMTSPs. Moreover, the proposed algorithm uses a new environmental selection operator to maintain a good balance between the objective space diversity and decision space diversity. We compare our algorithm with five state-of-the-art designs. Experimental results convincingly show that our algorithm is powerful in solving MMTSPs.

Evolutionary multimodal multiobjective optimization guided by growing neural gas

Evolutionary multimodal multiobjective optimization aims to search for a set of Pareto optimal solutions that are well distributed in both the objective and decision spaces. In recent years, there has been a growing interest in enhancing the search ability of evolutionary algorithms using machine learning techniques. However, there are few studies on machine learning-assisted evolutionary multimodal multiobjective optimization. In this paper, we propose an evolutionary multimodal multiobjective algorithm guided by growing neural gas, which learns the topological structure of the obtained good solutions during the evolution process. These good solutions are well-converged and well-distributed in the decision space, which indicate the shape of the Pareto optimal solution set. The proposed algorithm employs three novel mating selection operators based on the topological structure to efficiently and effectively search for Pareto optimal solutions. The first operator exploits Pareto optimal solutions within the topological structure, while the others explore Pareto optimal solutions around and far away from the topological structure, respectively. The proposed algorithm is compared with eight state-of-the-art evolutionary multimodal multiobjective algorithms on 20 test problems. Experimental results demonstrate that the proposed algorithm is competitive in solving these problems.

DBCC2: an improved difficulty-based cooperative co-evolution for many-modal optimization

Evolutionary multimodal optimization algorithms aim to provide multiple solutions simultaneously. Many studies have been conducted to design effective evolutionary algorithms for solving multimodal optimization problems. However, optimization problems with many global and acceptable local optima have not received much attention. This type of problem is undoubtedly challenging. In this study, we focus on problems with many optima, the so-called many-modal optimization problems, and this study is an extension of our previous conference work. First, a test suite including additively nonseparable many-modal optimization problems and partially additively separable many-modal optimization problems is designed. Second, an improved difficulty-based cooperative co-evolution algorithm (DBCC2) is proposed, which dynamically estimates the difficulties of subproblems and allocates the computational resources during the search. Experimental results show that DBCC2 has competitive performance.

An alternative way of evolutionary multimodal optimization: density-based population initialization strategy

Evolutionary algorithms rely on the population initialization strategy to determine a set of candidate solutions, which provide the preliminary knowledge of the problem landscape for the subsequent evolutionary process. However, the distribution of the initial population is rarely concerned in the multimodal evolutionary community. Moreover, a large initial population has not been attractive enough for evolutionary multimodal optimization, because it is difficult and challenging to achieve a balance between diversity and convergence within limited computing resources. As an extended version of our previous conference paper, this paper focuses on the construction of a large initial population that is beneficial for evolutionary multimodal optimization. First, we propose an improved density-based population initialization strategy to generate a uniform initial population with a certain degree of randomness. Then, after acquiring the raw fitness landscape of the problem, a fitness-weighted density-based population initialization strategy is proposed to explore potential global peaks. Finally, we design a multi-species cooperative coevolution algorithm for multimodal optimization, which balances the exploration and exploitation of such a population in the evolutionary process. Experimental results demonstrate that the proposed algorithm is better than the mainstream algorithms on the CEC2013 multimodal optimization benchmark.

Fuzzy K-means clustering with fast density peak clustering on multivariate kernel estimator with evolutionary multimodal optimization clusters on a large dataset

Many conventional optimization approaches concentrate more on addressing only one appropriate solution. Thus, these methods were to be utilized often, hence there were no chances of producing the intended solution. Therefore, the issue of multimodal optimization has to be considered. So, to reduce the difficulties by the clustering and further, it followed by the optimization technique. Here, the variety of real-time and artificial techniques are used. Using the FCDP-Fast Clustering with Density Peak, we calculate the density values after determining the center with the help of objective function. Then, the fuzzy clustering is applied to form the clustered groups with the density and center values. Finally, we optimize the data using the CDE-Crowding Differential Evaluation methodology. Performance analysis is then proceeded with some existing methods by using the performance metrics like NMI and ARI. After validation, it concluded that the proposed method was superior to the existing method.

System reliability analysis with small failure probability based on active learning Kriging model and multimodal adaptive importance sampling

System reliability analysis with small failure probability is investigated in this paper. Because multiple failure modes exist, the system performance function has multiple failure regions and multiple most probable points (MPPs). This paper reports an innovative method combining active learning Kriging (ALK) model with multimodal adaptive important sampling (MAIS). In each iteration of the proposed method, MPPs on a so-called surrogate limit state surface (LSS) of the system are explored, important samples are generated, optimal training points are chosen, the Kriging models are updated, and the surrogate LSS is refined. After several iterations, the surrogate LSS will converge to the true LSS. A recently proposed evolutionary multimodal optimization algorithm is adapted to obtain all the potential MPPs on the surrogate LSS, and a filtering technique is introduced to exclude improper solutions. In this way, the unbiasedness of our method is guaranteed. To avoid approximating the unimportant components, the training points are only chosen from the important samples located in the truncated candidate region (TCR). The proposed method is termed as ALK-MAIS-TCR. The accuracy and efficiency of ALK-MAIS-TCR are demonstrated by four complicated case studies.

Evolutionary Multimodal Optimization Based on Bi-Population and Multi-Mutation Differential Evolution

Evolutionary Multimodal Optimization Based on Bi-Population and Multi-Mutation Differential Evolution

Parameter-Free Voronoi Neighborhood for Evolutionary Multimodal Optimization

Neighborhood information plays an important role in improving the performance of evolutionary computation in various optimization scenarios, particularly in the context of multimodal optimization. Several neighborhood concepts, i.e., index-based neighborhood, nearest neighborhood, and fuzzy neighborhood, have been studied and engaged in the design of niching methods. However, the use of these neighborhood concepts requires the specification of some problem-related parameters, which is difficult to determine without a prior knowledge. In this paper, we introduce a new neighborhood concept based on a geometrical construction called Voronoi diagram. The new concept offers two advantages at the expense of increasing the computational complexity to a higher level. It eliminates the need of additional parameters and it is more informative than the existing ones. The information provided by the Voronoi neighbors of an individual can be exploited to estimate the evolutionary state. Based on the information, we divide the population into three groups and assign each group a different reproduction strategy to support the exploration and exploitation of the search space. We show the use of the concept in the design of an effective evolutionary algorithm for multimodal optimization. The experiments have been conducted to investigate the performance of the algorithm. The results reveal that the proposed algorithm compare favorably with the state-of-the-art algorithms designed based on other types of neighborhood concepts.

A Review of Evolutionary Multimodal Multiobjective Optimization

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.

Handling Imbalance Between Convergence and Diversity in the Decision Space in Evolutionary Multi-Modal Multi-Objective Optimization

There may exist more than one Pareto optimal solution with the same objective vector to a multimodal multiobjective optimization problem (MMOP). The difficulties in finding such solutions can be different. Although a number of evolutionary multimodal multiobjective algorithms (EMMAs) have been proposed, they are unable to solve such an MMOP due to their convergence-first selection criteria. They quickly converge to the Pareto optimal solutions which are easy to find and therefore lose diversity in the decision space. That is, such an MMOP features an imbalance between achieving convergence and preserving diversity in the decision space. In this article, we first present a set of imbalanced distance minimization benchmark problems. Then we propose an evolutionary algorithm using a convergence-penalized density method (CPDEA). In CPDEA, the distances among solutions in the decision space are transformed based on their local convergence quality. Their density values are estimated based on the transformed distances and used as the selection criterion. We compare CPDEA with five state-of-the-art EMMAs on the proposed benchmarks. Our experimental results show that CPDEA is clearly superior in solving these problems.

Learning Multimodal Parameters: A Bare-Bones Niching Differential Evolution Approach.

Most learning methods contain optimization as a substep, where the nondifferentiability and multimodality of objectives push forward the interplay of evolutionary optimization algorithms and machine learning models. The recently emerged evolutionary multimodal optimization (MMOP) technique enables the learning of diverse sets of effective parameters for the models simultaneously, providing new opportunities to the applications requiring both accuracy and diversity, such as ensemble, interactive, and interpretive learning. Targeting at locating multiple optima simultaneously in the multimodal landscape, this paper develops an efficient neighborhood-based niching algorithm. Bare-bones differential evolution is used as the baseline. Further, using Gaussian mutation with local mean and standard deviations, the neighborhoods capture niches that match well with the contours of peaks in the landscape. To increase diversity and enhance global exploration, the proposed algorithm embeds a diversity preserving operator to reinitialize converged or overlapped neighborhoods. The experimental results verify that the proposed algorithm has superior and consistent performance for a wide range of MMOP problems. Further, the algorithm has been successfully applied to train neural network ensembles, which validates its effectiveness and benefits of learning multimodal parameters.

A Novel Multimodal Optimization Algorithm for the Design of Electromagnetic Machines

The design of an electromagnetic machine is a nonlinear, multivariable, and multimodal optimization problem that incurs a great deal of computation time when calculating electromagnetic fields. To overcome these problems effectively, this paper proposes a new evolutionary multimodal optimization algorithm based on the Big Bang-Big Crunch method and aided by a surrogate model using the theory of compressed sensing. Its efficiency is demonstrated by assessing the optimization results for test functions. Moreover, to evaluate the feasibility of its application to an electromagnetic problem, an interior permanent magnet motor is designed using the proposed algorithm. The obtained results confirm that the proposed method has the ability to search for multiple optimal solutions with a good computational efficiency by reducing the number of fitness function evaluations during the optimization process.

Inducing Niching Behavior in Differential Evolution Through Local Information Sharing

In practical situations, it is very often desirable to detect multiple optimally sustainable solutions of an optimization problem. The population-based evolutionary multimodal optimization algorithms can be very helpful in such cases. They detect and maintain multiple optimal solutions during the run by incorporating specialized niching operations to aid the parallel localized convergence of population members around different basins of attraction. This paper presents an improved information-sharing mechanism among the individuals of an evolutionary algorithm for inducing efficient niching behavior. The mechanism can be integrated with stochastic real-parameter optimizers relying on differential perturbation of the individuals (candidate solutions) based on the population distribution. Various real-coded genetic algorithms (GAs), particle swarm optimization (PSO), and differential evolution (DE) fit the example of such algorithms. The main problem arising from differential perturbation is the unequal attraction toward the different basins of attraction that is detrimental to the objective of parallel convergence to multiple basins of attraction. We present our study through DE algorithm owing to its highly random nature of mutation and show how population diversity is preserved by modifying the basic perturbation (mutation) scheme through the use of random individuals selected probabilistically. By integrating the proposed technique with DE framework, we present three improved versions of well-known DE-based niching methods. Through an extensive experimental analysis, a statistically significant improvement in the overall performance has been observed upon integrating of our technique with the DE-based niching methods.

An Improved Parent-Centric Mutation With Normalized Neighborhoods for Inducing Niching Behavior in Differential Evolution

In real life, we often need to find multiple optimally sustainable solutions of an optimization problem. Evolutionary multimodal optimization algorithms can be very helpful in such cases. They detect and maintain multiple optimal solutions during the run by incorporating specialized niching operations in their actual framework. Differential evolution (DE) is a powerful evolutionary algorithm (EA) well-known for its ability and efficiency as a single peak global optimizer for continuous spaces. This article suggests a niching scheme integrated with DE for achieving a stable and efficient niching behavior by combining the newly proposed parent-centric mutation operator with synchronous crowding replacement rule. The proposed approach is designed by considering the difficulties associated with the problem dependent niching parameters (like niche radius) and does not make use of such control parameter. The mutation operator helps to maintain the population diversity at an optimum level by using well-defined local neighborhoods. Based on a comparative study involving 13 well-known state-of-the-art niching EAs tested on an extensive collection of benchmarks, we observe a consistent statistical superiority enjoyed by our proposed niching algorithm.

A GPU implementation of a hybrid evolutionary algorithm: GPuEGO

The high computation requirements of global optimization algorithms, when used to solve real optimization problems, have caused the appearance of different parallelization strategies using several parallel computing architectures. In this work, the Universal Evolutionary Global Optimizer is implemented in CUDA to be run on GPU architectures (GPuEGO). This parallelization of the referred evolutionary multimodal optimization algorithm is rather different from other previous parallel implementations designed to be executed into shared or distributed memory processors. In this case, due to the special characteristics of a GPU architecture, the original data structures are not valid and it has been necessary to redefine them and all the functions that operate with them. When this approach is applied the acceleration factors achieved by GPuEGO range from $${\times }$$ × 6.33 to $${\times }$$ × 23.20 depending on the test function.

Niching particle swarm optimization with local search for multi-modal optimization

Multimodal optimization is still one of the most challenging tasks for evolutionary computation. In recent years, many evolutionary multi-modal optimization algorithms have been developed. All these algorithms must tackle two issues in order to successfully solve a multi-modal problem: how to identify multiple global/local optima and how to maintain the identified optima till the end of the search. For most of the multi-modal optimization algorithms, the fine-local search capabilities are not effective. If the required accuracy is high, these algorithms fail to find the desired optima even after converging near them. To overcome this problem, this paper integrates a novel local search technique with some existing PSO based multimodal optimization algorithms to enhance their local search ability. The algorithms are tested on 14 commonly used multi-modal optimization problems and the experimental results suggest that the proposed technique not only increases the probability of finding both global and local optima but also reduces the average number of function evaluations.

Evolutionary multimodal optimization using the principle of locality

The principle of locality is one of the most widely used concepts in designing computing systems. To explore the principle in evolutionary computation, crowding differential evolution is incorporated with locality for multimodal optimization. Instead of generating trial vectors randomly, the first method proposed takes advantage of spatial locality to generate trial vectors. Temporal locality is also adopted to help generate offspring in the second method proposed. Temporal and spatial locality are then applied together in the third method proposed. Numerical experiments are conducted to compare the proposed methods with the state-of-the-art methods on benchmark functions. Experimental analysis is undertaken to observe the effect of locality and the synergy between temporal locality and spatial locality. Further experiments are also conducted on two application problems. One is the varied-line-spacing holographic grating design problem, while the other is the protein structure prediction problem. The numerical results demonstrate the effectiveness of the methods proposed.

Real-parameter evolutionary multimodal optimization — A survey of the state-of-the-art

Multimodal optimization amounts to finding multiple global and local optima (as opposed to a single solution) of a function, so that the user can have a better knowledge about different optimal solutions in the search space and as and when needed, the current solution may be switched to another suitable one while still maintaining the optimal system performance. Evolutionary Algorithms (EAs), due to their population-based approaches, are able to detect multiple solutions within a population in a single simulation run and have a clear advantage over the classical optimization techniques, which need multiple restarts and multiple runs in the hope that a different solution may be discovered every run, with no guarantee however. Numerous evolutionary optimization techniques have been developed since late 1970s for locating multiple optima (global or local). These techniques are commonly referred to as “niching” methods. Niching can be incorporated into a standard EA to promote and maintain formation of multiple stable subpopulations within a single population, with an aim to locate multiple globally optimal or suboptimal solutions simultaneously. This article is the first of its kind to present a comprehensive review of the basic concepts related to real-parameter evolutionary multimodal optimization, a survey of the major niching techniques, a detailed account of the adaptation of EAs from diverse paradigms to tackle multimodal problems, benchmark problems and performance measures.

Evolutionary swarm cooperative optimization in dynamic environments

A hybrid approach called Evolutionary Swarm Cooperative Algorithm (ESCA) based on the collaboration between a particle swarm optimization algorithm and an evolutionary algorithm is presented. ESCA is designed to deal with moving optima of optimization problems in dynamic environments. ESCA uses three populations of individuals: two EA populations and one Particle Swarm Population. The EA populations evolve by the rules of an evolutionary multimodal optimization algorithm being used to maintain the diversity of the search. The particle swarm confers precision to the search process. The efficiency of ESCA is evaluated by means of numerical experiments.