A better balance in metaheuristic algorithms: Does it exist?

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Abstract

The constant development of new metaheuristic algorithms has led to a saturation in the field of stochastic search. There are now hundreds of different algorithms that can be used to solve any problem. To produce a good performance, every metaheuristic method needs to address a satisfactory equilibrium between exploration and exploitation of the search space. Although exploration and exploitation represent two fundamental concepts in metaheuristics, the main questions about their combination and balance have not been yet completely understood. Most of the existent analyzes conducted on metaheuristic techniques consider only the comparison of their final results which cannot evaluate the nature of a good or bad balance. This paper presents an experimental analysis that quantitatively evaluates the balance between exploration and exploitation of several of the most important and better-known metaheuristic algorithms. In the study, a dimension-wise diversity measurement is used to assess the balance of each scheme considering a representative set of 42 benchmark problems that involve multimodal, unimodal, composite and shifted functions. As a result, the analysis provides several observations that allow understanding how this balance affects the results in each type of functions, and which balance is producing better solutions.

Introduction

In recent years, metaheuristics search algorithms have gained popularity as tools for solving a wide array of optimization problems in many different areas of application, including engineering design, digital image processing, and computer vision, networks and communications, power, and energy management, data analysis and machine learning, robotics, medical diagnosis, and others [1].

Most metaheuristic methods model population-based search schemes, in which a population of search agents (or individuals) applies specific sets of rules to explore different candidate solutions within a feasible solution space. These optimization frameworks present several advantages, including the interaction among individuals (which promotes the exchange of knowledge among different solutions) and the diversification of the population (which is important to ensure the efficient exploration of the search space and the ability to overcome local optima) [2].

Metaheuristic search methods are so numerous and varied in terms of design and potential applications [41]; however, for such an abundant family of optimization techniques, there seems to be a question that needs to be answered: Which part of the design in a metaheuristic algorithm contributes more to its performance? One widely accepted principle among researchers considers that metaheuristic search methods can reach a better performance when an appropriate balance between exploration and exploitation of solutions is achieved [3]. While there seems to exist a general agreement on this concept, in fact, there is barely a vague conception of what the balance of exploration and exploitation really represent [4,5]. Indeed, the classification of search operators and strategies present in a metaheuristic method is often ambiguous, since they can contribute in some way to explore or exploit the search space [6].

In the absence of consistent knowledge about the mechanism that controls the balance between exploration and exploitation, several attempts have been conducted to fill these gaps. Most of these efforts have proposed interesting metrics that allow quantifying the level of exploration and exploitation in search algorithms through the monitoring of the current population diversity [[4], [5], [6], [7], [8], [9], [10], [11]]. Although several indexes exist and more are being proposed, there is no definitive way to objectively measure the rate of exploration/exploitation provided in a metaheuristic scheme.

One of these metrics is the dimension-wise diversity measurement proposed in Ref. [10,12]. This index calculates the averaged sum distance between all the solutions to the median of each dimension and solutions. Then, the exploration percent of each iteration is calculated by dividing the diversity value between the maximum diversity value encounter in the whole optimization process. On the other hand, the exploitation rate is considered the inverse of exploration percent. This measurement allows to find out how spread or clustered the search agents are over specific iterations during the search process. These values give information on how much time the algorithm behaves exploring or exploiting solutions. In spite of the information provided by this index, it has not been adopted in the community yet to characterize the balance between exploration and exploitation in metaheuristic methods.

In this paper, an experimental analysis is proposed to quantitatively evaluate the balance between exploration and exploitation on several of the most important and better-known metaheuristic algorithms. As a result, the analysis provides several observations that allow understanding of how this balance affects the results in each type of functions, and which balance is producing for better solutions. The methods studied in this work include Artificial Bee Colony (ABC), Bat Algorithm (BA), Covariance Matrix Adaptation Evolution Strategies (CMA-ES), Crow Search Algorithm (CSA), Differential Evolution (DE), Firefly Algorithm (FA), Grey Wolf Optimizer (GWO), Moth-Flame Optimization (MFO), Particle Swarm Optimization (PSO), Social Spiders Optimization (SSO), Teaching-Learning Based Optimization (TLBO) and Whale Optimization Algorithm (WOA), which are considered some of the most important metaheuristic search algorithms on the current literature in virtue of their performance, novelty and potential.

This paper is organized as follows: In Section 2, we open a discussion related to the concepts of Exploration and Exploitation. In Section 3, we discuss the dimension-wide diversity measurement proposed in Ref. [12] and their applications for the analysis of exploration and exploitation in metaheuristic search methods. In Section 4, we present our experimental analysis. Section 5 presents a discussion of the results. Finally, in Section 6, conclusions are drawn.

Section snippets

Exploration and exploitation

For every metaheuristic algorithm, exploration and exploitation represent the most important characteristics for attaining success when solving a particular optimization problem. Exploration refers to the ability of a search algorithm to discover a diverse assortment of solutions, spread within different regions of the search space. On the other hand, exploitation emphasizes the idea of intensifying the search process over-promising regions of the solution space with the aim of finding better

Evaluation of the balance

Metaheuristic algorithms use a group of candidate solutions to explore the search space with the objective to find satisfactory solutions for an optimization problem. In general, search agents with the best solutions tend to direct the search process towards them. As a consequence of this attraction, the distance among search agents decreases while the effect of exploitation increases. On the other hand, when the distance among search agents increases, the effect of the exploration process is

Experiments

To analyze the balance between exploration and exploitation in many of the most popular and important metaheuristic algorithms, we examined the performance of 12 state-of-the-art optimization techniques: Artificial Bee Colony (ABC) [22,23], Bat Algorithm (BA) [24], Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [25], Crow Search Algorithm (CSA) [26], Differential Evolution (DE) [16], Firefly Algorithm (FA) [27], Grey Wolf Optimization (GWO) [19], Social Spiders Optimization (SSO) [28

Results analysis

According to this analysis, several interesting patterns have been identified. In the study, the best results have been obtained by WOA, CMA-ES, GWO and TLBO through an exploitation rate of over 90% (XPL%=10, XPT%=90). In general terms, from all schemes, the WOA maintains the best performance indexes.

From the experiments, it is clear that the best performances are obtained when the balances maintain a response of XPL%=10 and XPT%=90. In order to produce this behavior, the exploration process

Conclusions and future work

In this paper, an empirical evaluation of the balance between exploration and exploitation on metaheuristic algorithms has been conducted. In the study, a dimension-wise diversity measurement is used to assess the balance of each scheme considering a representative set of 42 benchmark problems that involve multimodal, unimodal, composite and shifted functions. In the majority of the 42 functions (multimodal, unimodal, hybrid and shifted) the balance that produced the best results was above 90%

CRediT authorship contribution statement

Bernardo Morales-Castañeda: Conceptualization, Investigation, Writing - original draft. Daniel Zaldívar: Resources, Data curation. Erik Cuevas: Resources, Writing - review & editing. Fernando Fausto: Visualization, Methodology. Alma Rodríguez: Software, Validation.

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