Display Omitted Chaotic maps have been embedded into Gravitational Search Algorithms (GSA) for the first time.The problem of trapping in local minima in GSA has been improved by the chaotic maps.The convergence rate of GSA has been improved.The statistical test allowed us to judge about the significance of the results.An adaptive normalization is proposed to smoothly transit from the exploration phase to the exploitation phase. In a population-based meta-heuristic, the search process is divided into two main phases: exploration versus exploitation. In the exploration phase, a random behavior is fruitful to explore the search space as extensive as possible. In contrast, a fast exploitation toward the promising regions is the main objective of the latter phase. It is really challenging to find a proper balance between these two phases because of the stochastic nature of population-based meta-heuristic algorithms. The literature shows that chaotic maps are able to improve both phases. This work embeds ten chaotic maps into the gravitational constant (G) of the recently proposed population-based meta-heuristic algorithm called Gravitational Search Algorithm (GSA). Also, an adaptive normalization method is proposed to transit from the exploration phase to the exploitation phase smoothly. As case studies, twelve shifted and biased benchmark functions evaluate the performance of the proposed chaos-based GSA algorithms in terms of exploration and exploitation. A statistical test called Wilcoxon rank-sum is done to judge about the significance of the results as well. The results demonstrate that sinusoidal map is the best map for improving the performance of GSA significantly.

Chaotic gravitational constants for the gravitational search algorithm

Population size of Differential Evolution (DE) algorithms is often specified by user and remains fixed during run. During the first decade since the introduction of DE the opinion that its population size should be related to the problem dimensionality prevailed, later the approaches to DE population size setting diversified. In large number of recently introduced DE algorithms the population size is considered to be problem-independent and often fixed to 100 or 50 individuals, but alongside a number of DE variants with flexible population size have been proposed. The present paper briefly reviews the opinions regarding DE population size setting and verifies the impact of the population size on the performance of DE algorithms. Ten DE algorithms with fixed population size, each with at least five different population size settings, and four DE algorithms with flexible population size are tested on CEC2005 benchmarks and CEC2011 real-world problems. It is found that the inappropriate choice of the population size may severely hamper the performance of each DE algorithm. Although the best choice of the population size depends on the specific algorithm, number of allowed function calls and problem to be solved, some rough guidelines may be sketched. When the maximum number of function calls is set to classical values, i.e. those specified for CEC2005 and CEC2011 competitions, for low-dimensional problems (with dimensionality below 30) the population size equal to 100 individuals is suggested; population sizes smaller than 50 are rarely advised. For higher-dimensional artificial problems the population size should often depend on the problem dimensionality d and be set to 3d–5d. Unfortunately, setting proper population size for higher-dimensional real-world problems (d>40) turns out too problem and algorithm-dependent to give any general guide; 200 individuals may be a first guess, but many DE approaches would need a much different choice, ranging from 50 to 10d. However, quite clear relation between the population size and the convergence speed has been found, showing that the fewer function calls are available, the lower population sizes perform better. Based on the extensive experimental results the use of adaptive population size is highly recommended, especially for higher-dimensional and real-world problems. However, which specific algorithms with population size adaptation perform better depends on the number of function calls allowed.

Review of Differential Evolution population size

Differential evolution (DE) is one of the best evolutionary algorithms (EAs). The effort of improving its performance has received great research attentions, such as adaptive DE (JADE). Based on the analysis on the aspects that may improve the performance of JADE, we introduce a modified JADE version with sorting crossover rate ( CR ). In JADE, CR values are generated based on mean value and Gaussian distribution. In the proposed algorithm, a smaller CR value is assigned to individual with better fitness value. Therefore, the components of the individuals, which have better fitness values, can appear in the offspring with higher possibility. In addition, the better offspring generated from last iteration are supposed to have better schemes, hence these schemes are preserved in next offspring generation procedure. This modified version is called as JADE algorithm with sorting CR (JADE_sort). The experiments results with several excellent algorithms show the effectiveness of JADE_sort.

Adaptive Differential Evolution With Sorting Crossover Rate for Continuous Optimization Problems

Self-adaptive parameter and strategy based particle swarm optimization for large-scale feature selection problems with multiple classifiers

Differential evolution (DE) is a popular evolutionary algorithm inspired by Darwin’s theory of evolution and has been studied extensively to solve different areas of optimisation and engineering applications since its introduction by Storn in 1997. This study aims to review the massive progress of DE in the research community by analysing the 192 articles published on this subject from 1997 to 2021, particularly studies in the past five years. The methodology used to search for relevant DE papers and an overview of the original DE are firstly explained. Recent advances in the modifications proposed to enhance the effectiveness and efficiency of the original DE are reviewed by analysing the strengths and weaknesses of each published work, followed by the potential applications of these DE variants in solving different real-world engineering problems. In contrast to most existing DE review papers, additional analyses are performed in this survey by investigating the impacts of various parameter settings on given DE variants to identify their optimal values required for solving certain problem classes. The qualities of modifications incorporated into selected DE variants are also evaluated by measuring the performance gains achieved in terms of search accuracy and/or efficiency against the original DE. The additional surveys conducted in this study are anticipated to provide more insightful perspectives for both beginners and experts of DE research, enabling their better understanding about current research trends and new motivations to outline appropriate strategic planning for future development works.

Differential evolution: A recent review based on state-of-the-art works

In DMPSADE, control parameters and mutation strategies could be automatically adjusted.We first proposed a new encoding for parameter control in DE algorithm.Roulette wheel is used to implement the selection of mutation strategies. Generally, the optimization problem has different relationships (i.e., linear, approximately linear, non-linear, or highly non-linear) with different optimized variables. The choices of control parameters and mutation strategies would directly affect the performance of differential evolution (DE) algorithm in satisfying the evolution requirement of each optimized variable and balancing its exploitation and exploration capabilities. Therefore, a self-adaptive DE algorithm with discrete mutation control parameters (DMPSADE) is proposed. In DMPSADE, each variable of each individual has its own mutation control parameter, and each individual has its own crossover control parameter and mutation strategy. DMPSADE was compared with 8 state-of-the-art DE variants and 3 non-DE algorithms by using 25 benchmark functions. The statistical results indicate that the average performance of DMPSADE is better than those of all other competitors.

Qinqin Fan

Papers

Self-adaptive differential evolution algorithm with discrete mutation control parameters