Differential Evolution Algorithm With Strategy Adaptation for Global Numerical Optimization

Differential evolution (DE) is an efficient and powerful population-based stochastic search technique for solving optimization problems over continuous space, which has been widely applied in many scientific and engineering fields. However, the success of DE in solving a specific problem crucially depends on appropriately choosing trial vector generation strategies and their associated control parameter values. Employing a trial-and-error scheme to search for the most suitable strategy and its associated parameter settings requires high computational costs. Moreover, at different stages of evolution, different strategies coupled with different parameter settings may be required in order to achieve the best performance. In this paper, we propose a self-adaptive DE (SaDE) algorithm, in which both trial vector generation strategies and their associated control parameter values are gradually self-adapted by learning from their previous experiences in generating promising solutions. Consequently, a more suitable generation strategy along with its parameter settings can be determined adaptively to match different phases of the search process/evolution. The performance of the SaDE algorithm is extensively evaluated (using codes available from P. N. Suganthan) on a suite of 26 bound-constrained numerical optimization problems and compares favorably with the conventional DE and several state-of-the-art parameter adaptive DE variants.     

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.     

A novel adaptive control strategy for decomposition-based multiobjective algorithm

Recently, evolutionary algorithm based on decomposition (MOEA/D) has been found to be very effective and efficient for solving complicated multiobjective optimization problems (MOPs). However, the selected differential evolution (DE) strategies and their parameter settings impact a lot on the performance of MOEA/D when tackling various kinds of MOPs. Therefore, in this paper, a novel adaptive control strategy is designed for a recently proposed MOEA/D with stable matching model, in which multiple DE strategies coupled with the parameter settings are adaptively conducted at different evolutionary stages and thus their advantages can be combined to further enhance the performance. By exploiting the historically successful experience, an execution probability is learned for each DE strategy to perform adaptive adjustment on the candidate solutions. The proposed adaptive strategies on operator selection and parameter settings are aimed at improving both of the convergence speed and population diversity, which are validated by our numerous experiments. When compared with several variants of MOEA/D such as MOEA/D, MOEA/D-DE, MOEA/D-DE+PSO, ENS-MOEA/D, MOEA/D-FRRMAB and MOEA/D-STM, our algorithm performs better on most of test problems.     

A novel mutation strategy selection mechanism for differential evolution based on local fitness landscape

The performance of differential evolution (DE) algorithm highly depends on the selection of mutation strategy. However, there are six commonly used mutation strategies in DE. Therefore, it is a challenging task to choose an appropriate mutation strategy for a specific optimization problem. For a better tackle this problem, in this paper, a novel DE algorithm based on local fitness landscape called LFLDE is proposed, in which the local fitness landscape information of the problem is investigated to guide the selection of the mutation strategy for each given problem at each generation. In addition, a novel control parameter adaptive mechanism is used to improve the proposed algorithm. In the experiments, a total of 29 test functions originated from CEC2017 single-objective test function suite which are utilized to evaluate the performance of the proposed algorithm. The Wilcoxon rank-sum test and Friedman rank test results reveal that the performance of the proposed algorithm is better than the other five representative DE algorithms.

     

DPD: An intelligent parallel hybrid algorithm for economic load dispatch problems with various practical constraints

The Economic Load Dispatch (ELD) problem has attracted much attention in the field of electric power system. This paper proposes a novel parallel hybrid optimization methodology aimed at solving ELD problem with various generator constraints. The proposed approach combines the Differential Evolution (DE) and Particle Swarm Optimization (PSO). Initially the whole population (in increasing order of fitness) is divided into three groups - Inferior Group, Mid Group and Superior Group. DE is employed in the inferior and superior groups, whereas PSO is used in the mid-group. The proposed method is called DPD as it uses DE-PSO-DE on a population in parallel manner. Two strategies namely Elitism (to retain the best obtained values so far) and Non-redundant search (to improve the solution quality) have been employed in DPD cycle. Moreover, the suitable mutation strategy for both DEs used in DPD is investigated over a set of 8 popular mutation strategies. Combination of 8 mutation strategies generated 64 different variants of DPD. Top 4 DPDs are investigated through IEEE CEC 2006 functions. Based on the performance analysis, best DPD is reported and further used in solving four different typical test systems of ELD problem. Numerical and graphical results indicate the efficiency, convergence characteristic and robustness of proposed DPD.     

Levy distributed parameter control in differential evolution for numerical optimization

Differential evolution (DE) algorithm is a population based stochastic search technique widely applied in scientific and engineering fields for global optimization over real parameter space. The performance of DE algorithm highly depends on the selection of values of the associated control parameters. Therefore, finding suitable values of control parameters is a challenging task and researchers have already proposed several adaptive and self-adaptive variants of DE. In the paper control parameters are adapted by levy distribution, named as Levy distributed DE (LdDE) which efficiently handles exploration and exploitation dilemma in the search space. In order to assure a fair comparison with existing parameter controlled DE algorithms, we apply the proposed method on number of well-known unimodal, basic and expanded multimodal and hybrid composite benchmark optimization functions having different dimensions. The empirical study shows that the proposed LdDE algorithm exhibits an overall better performance in terms of accuracy and convergence speed compared to five prominent adaptive DE algorithms.     

Adaptive direction information in differential evolution for numerical optimization

Differential evolution (DE) is a powerful evolutionary algorithm (EA) for numerical optimization. It has been successfully used in various scientific and engineering fields. In most of the DE algorithms, the neighborhood and direction information are not fully and simultaneously exploited to guide the search. Most recently, to make full use of these information, a DE framework with neighborhood and direction information (NDi-DE) was proposed. It was experimentally demonstrated that NDi-DE was effective for most of the DE algorithms. However, the performance of NDi-DE heavily depends on the selection of direction information. To alleviate this drawback and improve the performance of NDi-DE, the adaptive operator selection (AOS) mechanism is introduced into NDi-DE to adaptively select the direction information for the specific DE mutation strategy. Therefore, a new DE framework, adaptive direction information based NDi-DE (aNDi-DE), is proposed in this study. With AOS, the good balance between exploration and exploitation of aNDi-DE can be dynamically achieved. In order to evaluate the effectiveness of aNDi-DE, the proposed framework is applied to the original DE algorithms, as well as several advanced DE variants. Experimental results show that aNDi-DE is able to adaptively select the most suitable type of direction information for the specific DE mutation strategy during the evolutionary process. The efficiency and robustness of aNDi-DE are also confirmed by comparing with NDi-DE.     

差异演化算法应用问题研究

差异演化算法作为一种较新的演化算法,具有全局搜索能力强、控制参数少、算法简单、容易实现等特点,但在具体应用中仍面临着演化模式如何选择及最佳种群大小如何取值等问题。本文利用典型的五类多变量函数对差异演化算法常用演化模式进行了比较测试研究,给出了演化模式选择建议;在同时考虑种群大小对差异演化算法优化效率及优化结果两方面影响的情况下,利用测试函数对最佳种群大小的取值进行了研究,给出了差异演化算法用于多变量函数优化时种群大小的适用取值范围。     

DE/BBO: a hybrid differential evolution with biogeography-based optimization for global numerical optimization

Differential evolution (DE) is a fast and robust evolutionary algorithm for global optimization. It has been widely used in many areas. Biogeography-based optimization (BBO) is a new biogeography inspired algorithm. It mainly uses the biogeography-based migration operator to share the information among solutions. In this paper, we propose a hybrid DE with BBO, namely DE/BBO, for the global numerical optimization problem. DE/BBO combines the exploration of DE with the exploitation of BBO effectively, and hence it can generate the promising candidate solutions. To verify the performance of our proposed DE/BBO, 23 benchmark functions with a wide range of dimensions and diverse complexities are employed. Experimental results indicate that our approach is effective and efficient. Compared with other state-of-the-art DE approaches, DE/BBO performs better, or at least comparably, in terms of the quality of the final solutions and the convergence rate. In addition, the influence of the population size, dimensionality, different mutation schemes, and the self-adaptive control parameters of DE are also studied.     

An adaptive differential evolution algorithm with novel mutation and crossover strategies for global numerical optimization

Differential evolution (DE) is one of the most powerful stochastic real parameter optimizers of current interest. In this paper, we propose a new mutation strategy, a fitness-induced parent selection scheme for the binomial crossover of DE, and a simple but effective scheme of adapting two of its most important control parameters with an objective of achieving improved performance. The new mutation operator, which we call DE/current-to-gr_best/1, is a variant of the classical DE/current-to-best/1 scheme. It uses the best of a group (whose size is q% of the population size) of randomly selected solutions from current generation to perturb the parent (target) vector, unlike DE/current-to-best/1 that always picks the best vector of the entire population to perturb the target vector. In our modified framework of recombination, a biased parent selection scheme has been incorporated by letting each mutant undergo the usual binomial crossover with one of the p top-ranked individuals from the current population and not with the target vector with the same index as used in all variants of DE. A DE variant obtained by integrating the proposed mutation, crossover, and parameter adaptation strategies with the classical DE framework (developed in 1995) is compared with two classical and four state-of-the-art adaptive DE variants over 25 standard numerical benchmarks taken from the IEEE Congress on Evolutionary Computation 2005 competition and special session on real parameter optimization. Our comparative study indicates that the proposed schemes improve the performance of DE by a large magnitude such that it becomes capable of enjoying statistical superiority over the state-of-the-art DE variants for a wide variety of test problems. Finally, we experimentally demonstrate that, if one or more of our proposed strategies are integrated with existing powerful DE variants such as jDE and JADE, their performances can also be enhanced.     

Modified differential evolution approach for optimization of planar parallel manipulators force capabilities

The global optimization problem is not easy to solve and is still an open challenge for researchers since an analytical optimal solution is difficult to obtain even for relatively simple application problems. Conventional deterministic numerical algorithms tend to stop the search in local minimum nearest to the input starting point, mainly when the optimization problem presents nonlinear, non-convex and non-differential functions, multimodal and nonlinear. Nowadays, the use of evolutionary algorithms (EAs) to solve optimization problems is a common practice due to their competitive performance on complex search spaces. EAs are well known for their ability to deal with nonlinear and complex optimization problems. The primary advantage of EAs over other numerical methods is that they just require the objective function values, while properties such as differentiability and continuity are not necessary. In this context, the differential evolution (DE), a paradigm of the evolutionary computation, has been widely used for solving numerical global optimization problems in continuous search space. DE is a powerful population-based stochastic direct search method. DE simulates natural evolution combined with a mechanism to generate multiple search directions based on the distribution of solutions in the current population. Among DE advantages are its simple structure, ease of use, speed, and robustness, which allows its application on several continuous nonlinear optimization problems. However, the performance of DE greatly depends on its control parameters, such as crossover rate, mutation factor, and population size and it often suffers from being trapped in local optima. Conventionally, users have to determine the parameters for problem at hand empirically. Recently, several adaptive variants of DE have been proposed. In this paper, a modified differential evolution (MDE) approach using generation-varying control parameters (mutation factor and crossover rate) is proposed and evaluated. The proposed MDE presents an efficient strategy to improve the search performance in preventing of premature convergence to local minima. The efficiency and feasibility of the proposed MDE approach is demonstrated on a force optimization problem in Robotics, where the force capabilities of a planar 3-RRR parallel manipulator are evaluated considering actuation limits and different assembly modes. Furthermore, some comparison results of MDE approach with classical DE to the mentioned force optimization problem are presented and discussed.     

Adaptive differential evolution algorithm with novel mutation strategies in multiple sub-populations

Differential evolution (DE) algorithm has been shown to be a very effective and efficient approach for solving global numerical optimization problems, which attracts a great attention of scientific researchers. Generally, most of DE algorithms only evolve one population by using certain kind of DE operators. However, as observed in nature, the working efficiency can be improved by using the concept of work specialization, in which the entire group should be divided into several sub-groups that are responsible for different tasks according to their capabilities. Inspired by this phenomenon, a novel adaptive multiple sub-populations based DE algorithm is designed in this paper, named MPADE, in which the parent population is split into three sub-populations based on the fitness values and then three novel DE strategies are respectively performed to take on the responsibility for either exploitation or exploration. Furthermore, a simple yet effective adaptive approach is designed for parameter adjustment in the three DE strategies and a replacement strategy is put forward to fully exploit the useful information from the trial vectors and target vectors, which enhance the optimization performance. In order to validate the effectiveness of MPADE, it is tested on 55 benchmark functions and 15 real world problems. When compared with other DE variants, MPADE performs better in most of benchmark problems and real-world problems. Moreover, the impacts of the MPADE components and their parameter sensitivity are also analyzed experimentally.     

Learning-enhanced differential evolution for numerical optimization

Differential evolution (DE) is a simple and powerful population-based search algorithm, successfully used in various scientific and engineering fields. However, DE is not free from the problems of stagnation and premature convergence. Hence, designing more effective search strategies to enhance the performance of DE is one of the most salient and active topics. This paper proposes a new method, called learning-enhanced DE (LeDE) that promotes individuals to exchange information systematically. Distinct from the existing DE variants, LeDE adopts a novel learning strategy, namely clustering-based learning strategy (CLS). In CLS, there are two levels of learning strategies, intra-cluster learning strategy and inter-cluster learning strategy. They are adopted for exchanging information within the same cluster and between different clusters, respectively. Experimental studies over 23 benchmark functions show that LeDE significantly outperforms the conventional DE. Compared with other clustering-based DE algorithms, LeDE can obtain better solutions. In addition, LeDE is also shown to be significantly better than or at least comparable to several state-of-art DE variants as well as some other evolutionary algorithms.     

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.     

On the selection of solutions for mutation in differential evolution

Differential evolution (DE) is a kind of evolutionary algorithms, which is suitable for solving complex optimization problems. Mutation is a crucial step in DE that generates new solutions from old ones. It was argued and has been commonly adopted in DE that the solutions selected for mutation should have mutually different indices. This restrained condition, however, has not been verified either theoretically or empirically yet. In this paper, we empirically investigate the selection of solutions for mutation in DE. From the observation of the extensive experiments, we suggest that the restrained condition could be relaxed for some classical DE versions as well as some advanced DE variants. Moreover, relaxing the restrained condition may also be useful in designing better future DE algorithms.     

A differential-based harmony search algorithm for the optimization of continuous problems

The performance of the Harmony Search (HS) algorithm is highly dependent on the parameter settings and the initialization of the Harmony Memory (HM). To address these issues, this paper presents a new variant of the HS algorithm, which is called the DH/best algorithm, for the optimization of globally continuous problems. The proposed DH/best algorithm introduces a new improvisation method that differs from the conventional HS in two respects. First, the random initialization of the HM is replaced with a new method that effectively initializes the harmonies and reduces randomness. Second, the conventional pitch adjustment method is replaced by a new pitch adjustment method that is inspired by a Differential Evolution (DE) mutation strategy known as DE/best/1. Two sets of experiments are performed to evaluate the proposed algorithm. In the first experiment, the DH/best algorithm is compared with other variants of HS based on 12 optimization functions. In the second experiment, the complete CEC2014 problem set is used to compare the performance of the DH/best algorithm with six well-known optimization algorithms from different families. The experimental results demonstrate the superiority of the proposed algorithm in convergence, precision, and robustness.     

Enhancing differential evolution with role assignment scheme

As one of the most popular evolutionary algorithms, differential evolution (DE) has been used for solving a wide range of real-world problems. The performance of DE highly depends on the chosen mutation strategy and control parameter settings. Although the conventional trial-and-error procedure can be used to elaborately select the proper strategy and to tune the parameter values, this procedure is often very time-consuming and is not suitable for practitioners without a priori experience. To tackle this problem, DE with a novel role assignment (RA) scheme is proposed in this paper. In the RA scheme, both the fitness information and positional information of individuals are utilized to dynamically divide the population into several groups. Each group is considered as a role, which has its own mutation strategy and parameter settings and is expected to play a different role in the evolution process. To verify the performance of our approach, experiments are conducted on 23 well-known benchmark functions. Results show that our approach is better than, or at least comparable to, several state-of-the-art DE variants.     

Optimization of fed-batch fermentation processes with bio-inspired algorithms

The optimization of the feeding trajectories in fed-batch fermentation processes is a complex problem that has gained attention given its significant economical impact. A number of bio-inspired algorithms have approached this task with considerable success, but systematic and statistically significant comparisons of the different alternatives are still lacking. In this paper, the performance of different metaheuristics, such as Evolutionary Algorithms (EAs), Differential Evolution (DE) and Particle Swarm Optimization (PSO) is compared, resorting to several case studies taken from literature and conducting a thorough statistical validation of the results. DE obtains the best overall performance, showing a consistent ability to find good solutions and presenting a good convergence speed, with the DE/rand variants being the ones with the best performance. A freely available computational application, OptFerm, is described that provides an interface allowing users to apply the proposed methods to their own models and data.     

Self-adaptive differential evolution with multi-trajectory search for large-scale optimization

In this paper, self-adaptive differential evolution (DE) is enhanced by incorporating the JADE mutation strategy and hybridized with modified multi-trajectory search (MMTS) algorithm (SaDE-MMTS) to solve large-scale continuous optimization problems. The JADE mutation strategy, the “DE/current-to-pbest” which is a variation of the classic “DE/current-to-best”, is used for generating mutant vectors. After the mutation phase, the binomial (uniform) crossover, the exponential crossover as well as no crossover option are used to generate each pair of target and trial vectors. By utilizing the self-adaptation in SaDE, both trial vector generation strategies and their associated control parameter values are gradually self-adapted by learning from their previous experiences in generating promising solutions. Consequently, suitable offspring generation strategy along with associated parameter settings will be determined adaptively to match different phases of the search process. MMTS is applied frequently to refine several diversely distributed solutions at different search stages satisfying both the global and the local search requirement. The initialization of step sizes is also defined by a self-adaption during every MMTS step. The success rates of both SaDE and the MMTS are determined and compared; consequently, future function evaluations for both search algorithms are assigned proportionally to their recent past performance. The proposed SaDE-MMTS is employed to solve the 19 numerical optimization problems in special issue of soft computing on scalability of evolutionary algorithms for large-scale continuous optimization problems and competitive results are presented.     

An improved Differential Evolution algorithm using learning automata and population topologies

A new variant of Differential Evolution (DE), called ADE-Grid, is presented in this paper which adapts the mutation strategy, crossover rate (CR) and scale factor (F) during the run. In ADE-Grid, learning automata (LA), which are powerful decision making machines, are used to determine the proper value of the parameters CR and F, and the suitable strategy for the construction of a mutant vector for each individual, adaptively. The proposed automata based DE is able to maintain the diversity among the individuals and encourage them to move toward several promising areas of the search space as well as the best found position. Numerical experiments are conducted on a set of twenty four well-known benchmark functions and one real-world engineering problem. The performance comparison between ADE-Grid and other state-of-the-art DE variants indicates that ADE-Grid is a viable approach for optimization. The results also show that the proposed ADE-Grid improves the performance of DE in terms of both convergence speed and quality of final solution.