Performance comparison of self-adaptive and adaptive differential evolution algorithms

Differential evolution (DE) has been shown to be a simple, yet powerful, evolutionary algorithm for global optimization for many real problems. Adaptation, especially self-adaptation, has been found to be highly beneficial for adjusting control parameters, especially when done without any user interaction. This paper presents differential evolution algorithms, which use different adaptive or self-adaptive mechanisms applied to the control parameters. Detailed performance comparisons of these algorithms on the benchmark functions are outlined.     

An improved differential evolution algorithm with fitness-based adaptation of the control parameters

Differential Evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms of current interest. DE operates through the similar computational steps as employed by a standard Evolutionary Algorithm (EA). However, unlike the traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used, which makes the scheme self-organizing in this respect. Scale Factor (F) and Crossover Rate (Cr) are two very important control parameters of DE since the former regulates the step-size taken while mutating a population member in DE and the latter controls the number of search variables inherited by an offspring from its parent during recombination. This article describes a very simple yet very much effective adaptation technique for tuning both F and Cr, on the run, without any user intervention. The adaptation strategy is based on the objective function value of individuals in the DE population. Comparison with the best-known and expensive variants of DE over fourteen well-known numerical benchmarks and one real-life engineering problem reflects the superiority of proposed parameter tuning scheme in terms of accuracy, convergence speed, and robustness.     

Performance improvement of self-adaptive evolutionary methods with a dynamic lower bound

Recent research on self-adaptive evolutionary programming (EP) methods evidenced the problem of premature convergence. Self-adaptive evolutionary programming methods converge prematurely because their object variables evolve more slowly than do their strategy parameters, which subsequently leads to a stagnation of object variables at a non-optimum value. To address this problem, a dynamic lower bound has been proposed, which is defined here as the differential step lower bound (DSLB) on the strategy parameters. The DSLB on an object variable depends on its absolute distance from the corresponding object variable of the best individual in the population pool. The performance of the self-adaptive EP algorithm with DSLB has been verified over eight different test functions of varied complexities.     

Population size reduction for the differential evolution algorithm

This paper studies the efficiency of a recently defined population-based direct global optimization method called Differential Evolution with self-adaptive control parameters. The original version uses fixed population size but a method for gradually reducing population size is proposed in this paper. It improves the efficiency and robustness of the algorithm and can be applied to any variant of a Differential Evolution algorithm. The proposed modification is tested on commonly used benchmark problems for unconstrained optimization and compared with other optimization methods such as Evolutionary Algorithms and Evolution Strategies.     

差异演化的实验研究

首先基于一些实例研究了差异演化(DE)的参数选择问题；然后在分析DE特点的基础上，将缩放因子F由固定数值设为随机函数，实现了一个简化的DE版本(SDE)，该方法不仅减少了需调整的参数，而且对CR的参数选择更为宽松．与已有文献中遗传算法的带约束型数值优化问题的实验结果对比，表明SDE能在较少的计算次数内获得较好的结果。     

A study on scale factor in distributed differential evolution

This paper proposes the employment of multiple scale factor values within distributed differential evolution structures. Four different scale factor schemes are proposed, tested, compared and analyzed. Two schemes simply employ multiple scale factor values and two also include an update logic during the evolution. The four schemes have been integrated for comparison within three recently proposed distributed differential evolution structures and tested on several various test problems.Numerical results show that, on average, the employment of multiple scale factors is beneficial since in most cases it leads to significant improvements in performance with respect to standard distributed algorithms. Although proper choice of a scale factor scheme appears to be dependent on the distributed structure, any of the proposed simple schemes has proven to significantly improve upon the single scale factor distributed differential evolution algorithms.     

Multi-objective differential evolution with performance-metric-based self-adaptive mutation operator for chemical and biochemical dynamic optimization problems

Each mutation operator of differential evolution (DE) algorithm is generally suitable for certain specific types of multi-objective optimization problems (MOPs) or particular stages of the evolution. To automatically select an appropriate mutation operator for solving MOPs in different phases of the evolution, a multi-objective differential evolution with performance-metric-based self-adaptive mutation operator (MODE-PMSMO) is proposed in this study. In MODE-PMSMO, a modified inverted generational distance (IGD) is utilized to evaluate the performance of each mutation operator and guide the evolution of mutation operators. The proposed MODE-PMSMO is then compared with seven multi-objective evolutionary algorithms (MOEAs) on five bi-objective and five tri-objective optimization problems. Generally, MODE-PMSMO exhibits the best average performance among all compared algorithms on ten MOPs. Additionally, MODE-PMSMO is employed to solve four typical multi-objective dynamic optimization problems in chemical and biochemical processes. Experimental results indicate that MODE-PMSMO is suitable for solving these actual problems and can provide a set of nondominated solutions for references of decision makers.     

A differential evolution algorithm with self-adapting strategy and control parameters

This paper presents a Differential Evolution algorithm with self-adaptive trial vector generation strategy and control parameters (SspDE) for global numerical optimization over continuous space. In the SspDE algorithm, each target individual has an associated strategy list (SL), a mutation scaling factor F list (FL), and a crossover rate CR list (CRL). During the evolution, a trial individual is generated by using a strategy, F, and CR taken from the lists associated with the target vector. If the obtained trial individual is better than the target vector, the used strategy, F, and CR will enter a winning strategy list (wSL), a winning F list (wFL), and a winning CR list (wCRL), respectively. After a given number of iterations, the FL, CRL or SL will be refilled at a high probability by selecting elements from wFL, wCRL and wSL or randomly generated values. In this way, both the trial vector generation strategy and its associated parameters can be gradually self-adapted to match different phases of evolution by learning from their previous successful experience. Extensive computational simulations and comparisons are carried out by employing a set of 19 benchmark problems from the literature. The computational results show that overall the SspDE algorithm performs better than the state-of-the-art differential evolution variants.     

Nonlinear system identification using memetic differential evolution trained neural networks

Several gradient-based approaches such as back propagation (BP) and Levenberg Marquardt (LM) methods have been developed for training the neural network (NN) based systems. But, for multimodal cost functions these procedures may lead to local minima, therefore, the evolutionary algorithms (EAs) based procedures are considered as promising alternatives. In this paper we focus on a memetic algorithm based approach for training the multilayer perceptron NN applied to nonlinear system identification. The proposed memetic algorithm is an alternative to gradient search methods, such as back-propagation and back-propagation with momentum which has inherent limitations of many local optima. Here we have proposed the identification of a nonlinear system using memetic differential evolution (DE) algorithm and compared the results with other six algorithms such as Back-propagation (BP), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), Genetic Algorithm Back-propagation (GABP), Particle Swarm Optimization combined with Back-propagation (PSOBP). In the proposed system identification scheme, we have exploited DE to be hybridized with the back propagation algorithm, i.e. differential evolution back-propagation (DEBP) where the local search BP algorithm is used as an operator to DE. These algorithms have been tested on a standard benchmark problem for nonlinear system identification to prove their efficacy. First examples shows the comparison of different algorithms which proves that the proposed DEBP is having better identification capability in comparison to other. In example 2 good behavior of the identification method is tested on an one degree of freedom (1DOF) experimental aerodynamic test rig, a twin rotor multi-input-multi-output system (TRMS), finally it is applied to Box and Jenkins Gas furnace benchmark identification problem and its efficacy has been tested through correlation analysis.     

An improved differential evolution algorithm with triangular mutation for global numerical optimization

This paper presents an Improved Differential Evolution (IDE) algorithm for solving global numerical optimization problems over continuous space. The proposed algorithm introduces a new triangular mutation rule based on the convex combination vector of the triplet defined by the three randomly chosen vectors and the difference vector between the best and the worst individuals among the three randomly selected vectors. The mutation rule is combined with the basic mutation strategy through a non-linear decreasing probability rule. Furthermore, a restart mechanism is also proposed to avoid premature convergence. IDE is tested on a well-known set of unconstrained problems and shows its superiority to state-of-the-art differential evolution variants.     

An adaptive differential evolution algorithm with an aging leader and challengers mechanism

An adaptive differential evolution algorithm with an aging leader and challengers mechanism, called ADE-ALC, is proposed to solve optimization problems. In ADE-ALC algorithm, the aging mechanism is introduced into the framework of differential evolution to maintain diversity of the population. The key control parameters are adaptively updated based on given probability distributions which could learn from their successful experiences to generate the promising parameters at the next generation. One of the two local search operators is randomly selected to generate challengers which are beneficial for increasing the diversity of population. Finally, the effectiveness of the ADE-ALC algorithm is verified by the numerical results of twenty-five benchmark test functions.     

Utilizing cumulative population distribution information in differential evolution

Differential evolution (DE) is one of the most popular paradigms of evolutionary algorithms. In general, DE does not exploit distribution information provided by the population and, as a result, its search performance is limited. In this paper, cumulative population distribution information of DE has been utilized to establish an Eigen coordinate system by making use of covariance matrix adaptation. The crossover operator of DE implemented in the Eigen coordinate system has the capability to identify the features of the fitness landscape. Furthermore, we propose a cumulative population distribution information based DE framework called CPI-DE. In CPI-DE, for each target vector, two trial vectors are generated based on both the original coordinate system and the Eigen coordinate system. Then, the target vector is compared with these two trial vectors and the best one will survive into the next generation. CPI-DE has been applied to two classic versions of DE and three state-of-the-art variants of DE for solving two sets of benchmark test functions, namely, 28 test functions with 30 and 50 dimensions at the 2013 IEEE Congress on Evolutionary Computation, and 30 test functions with 30 and 50 dimensions at the 2014 IEEE Congress on Evolutionary Computation. The experimental results suggest that CPI-DE is an effective framework to enhance the performance of DE.     

A self-adaptive differential evolution algorithm for binary CSPs

A novel self-adaptive differential evolution (SADE) algorithm is proposed in this paper. SADE adjusts the mutation rate F and the crossover rate CR adaptively, taking account of the different distribution of population. In order to balance an individual’s exploration and exploitation capability for different evolving phases, F and CR are equal to two different self-adjusted nonlinear functions. Attention is concentrated on varying F and CR dynamically with each generation evolution. SADE maintains the diversity of population and improves the global convergence ability. It also improves the efficiency and success rate and avoids the premature convergence. Simulation and comparisons based on test-sets of CSPs demonstrate the effectiveness, efficiency and robustness of the proposed algorithm.     

A hybrid differential evolution algorithm integrated with an ant system and its application

A new hybrid differential evolution algorithm, in which an ant system is used to select the optimal base vector of mutation operation, named the ant system differential evolution (ASDE), is proposed. In ASDE, each dimension in the feasible solution space is divided into several subspaces evenly, and each subspace is marked with the same initial intensity of pheromone trails. The probability of choosing an individual as the base vector is influenced by the visibility and pheromone quantity of the individual. The trail of the selected base vector’s location subspaces will be reinforced with some pheromones, when the offspring is better than its parent. The experimental results show that the ASDE generally outperforms the other differential evolution algorithms for nine benchmark functions. Furthermore, the ASDE is applied to develop the global kinetic model for SO₂ oxidation on the Cs-Rb-V catalyst, and satisfactory results are obtained.     

Differential evolution in constrained numerical optimization: An empirical study

Motivated by the recent success of diverse approaches based on differential evolution (DE) to solve constrained numerical optimization problems, in this paper, the performance of this novel evolutionary algorithm is evaluated. Three experiments are designed to study the behavior of different DE variants on a set of benchmark problems by using different performance measures proposed in the specialized literature. The first experiment analyzes the behavior of four DE variants in 24 test functions considering dimensionality and the type of constraints of the problem. The second experiment presents a more in-depth analysis on two DE variants by varying two parameters (the scale factor F and the population size NP), which control the convergence of the algorithm. From the results obtained, a simple but competitive combination of two DE variants is proposed and compared against state-of-the-art DE-based algorithms for constrained optimization in the third experiment. The study in this paper shows (1) important information about the behavior of DE in constrained search spaces and (2) the role of this knowledge in the correct combination of variants, based on their capabilities, to generate simple but competitive approaches.     

Mining significant fuzzy association rules with differential evolution algorithm

This article presents a new differential evolution (DE) algorithm for mining optimized statistically significant fuzzy association rules that are abundant in number and high in rule interestingness measure (RIM) values, with strict control over the risk of spurious rules. The risk control over spurious rules, as the most distinctive feature of the proposed DE compared with existing evolutionary algorithms (EAs) for association rule mining (ARM), is realized via two new statistically sound significance tests on the rules. The two tests, in the experimentwise and generationwise adjustment approach, can respectively limit the familywise error rate (the probability that any spurious rules occur in the ARM result) and percentage of spurious rules upon the user specified level. Experiments on variously sized data show that the proposed DE can keep the risk of spurious rules well below the user specified level, which is beyond the ability of existing EA-based ARM. The new method also carries forward the advantages of EA-based ARM and distinctive merits of DE in optimizing the rules: it can obtain several times as many rules and as high RIM values as conventional non-evolutionary ARM, and even more informative rules and better RIM values than genetic-algorithm-based ARM. Case studies on hotel room price determinants and wildfire risk factors demonstrate the practical usefulness of the proposed DE.     

An evolving surrogate model-based differential evolution algorithm

Differential evolution (DE) is a simple and effective approach for solving numerical optimization problems. However, the performance of DE is sensitive to the choice of mutation and crossover strategies and their associated control parameters. Therefore, to achieve optimal performance, a time-consuming parameter tuning process is required. In DE, the use of different mutation and crossover strategies with different parameter settings can be appropriate during different stages of the evolution. Therefore, to achieve optimal performance using DE, various adaptation, self-adaptation, and ensemble techniques have been proposed. Recently, a classification-assisted DE algorithm was proposed to overcome trial and error parameter tuning and efficiently solve computationally expensive problems. In this paper, we present an evolving surrogate model-based differential evolution (ESMDE) method, wherein a surrogate model constructed based on the population members of the current generation is used to assist the DE algorithm in order to generate competitive offspring using the appropriate parameter setting during different stages of the evolution. As the population evolves over generations, the surrogate model also evolves over the iterations and better represents the basin of search by the DE algorithm. The proposed method employs a simple Kriging model to construct the surrogate. The performance of ESMDE is evaluated on a set of 17 bound-constrained problems. The performance of the proposed algorithm is compared to state-of-the-art self-adaptive DE algorithms: the classification-assisted DE algorithm, regression-assisted DE algorithm, and ranking-assisted DE algorithm.     

Repairing the crossover rate in adaptive differential evolution

Differential evolution (DE) is a simple yet powerful evolutionary algorithm (EA) for global numerical optimization. However, its performance is significantly influenced by its parameters. Parameter adaptation has been proven to be an efficient way for the enhancement of the performance of the DE algorithm. Based on the analysis of the behavior of the crossover in DE, we find that the trial vector is directly related to its binary string, but not directly related to the crossover rate. Based on this inspiration, in this paper, we propose a crossover rate repair technique for the adaptive DE algorithms that are based on successful parameters. The crossover rate in DE is repaired by its corresponding binary string, i.e. by using the average number of components taken from the mutant. The average value of the binary string is used to replace the original crossover rate. To verify the effectiveness of the proposed technique, it is combined with an adaptive DE variant, JADE, which is a highly competitive DE variant. Experiments have been conducted on 25 functions presented in CEC-2005 competition. The results indicate that our proposed crossover rate technique is able to enhance the performance of JADE. In addition, compared with other DE variants and state-of-the-art EAs, the improved JADE method obtains better, or at least comparable, results in terms of the quality of final solutions and the convergence rate.     

差分进化算法研究进展

作为一种简单而有效的新兴计算技术,差分进化算法（DE）已受到学术界和工程界的广泛关注．并取得了许多成功应用．为此,围绕差分进化算法的原理、特点、改进及其应用等方面进行全面综述．重点介绍了针对复杂环境的差分进化算法研究内容,包括多目标、约束、离散和噪声环境下的优化等．最后提出了有待进一步研究的若干方向．     

Static force capability optimization of humanoids robots based on modified self-adaptive differential evolution

The current society requires solutions for many problems in safety, economy, and health. The social concerns on the high rate of repetitive strain injury, work-related osteomuscular disturbances, and domestic issues involving the elderly and handicapped are some examples. Therefore, studies on complex machines with structures similar to humans, known as humanoids robots, as well as emerging optimization metaheuristics have been increasing. The combination of these technologies may result in robust, safe, reliable, and flexible machines that can substitute humans in multiple tasks. In order to contribute to this topic, the static modeling of a humanoid robot and the optimization of its static force capability through a modified self-adaptive differential evolution (MSaDE) approach is proposed and evaluated in this study. Unlike the original SaDE, MSaDE employs a new combination of strategies and an adaptive scaling factor mechanism. In order to verify the effectiveness of the proposed MSaDE, a series of controlled experiments are performed. Moreover, some statistical tests are applied, an analysis of the results is carried out, and a comparative study of the MSaDE performance with other metaheuristics is presented. The results show that the proposed MSaDE is robust, and its performance is better than other powerful algorithms in the literature when applied to a humanoid robot model for the pushing and pulling tasks.