Differential Evolution with Adaptive Mutation and Parameter Control Using Lévy Probability Distribution

Differential evolution (DE) has become a very popular and effective global optimization algorithm in the area of evolutionary computation. In spite of many advantages such as conceptual simplicity, high efficiency and ease of use, DE has two main components, i.e., mutation scheme and parameter control, which significantly influence its performance. In this paper we intend to improve the performance of DE by using carefully considered strategies for both of the two components. We first design an adaptive mutation scheme, which adaptively makes use of the bias of superior individuals when generating new solutions. Although introducing such a bias is not a new idea, existing methods often use heuristic rules to control the bias. They can hardly maintain the appropriate balance between exploration and exploitation during the search process, because the preferred bias is often problem and evolution-stage dependent. Instead of using any fixed rule, a novel strategy is adopted in the new adaptive mutation scheme to adjust the bias dynamically based on the identified local fitness landscape captured by the current population. As for the other component, i.e., parameter control, we propose a mechanism by using the Lvy probability distribution to adaptively control the scale factor F of DE. For every mutation in each generation, an F i is produced from one of four different Lvy distributions according to their historical performance. With the adaptive mutation scheme and parameter control using Lvy distribution as the main components, we present a new DE variant called Lvy DE (LDE). Experimental studies were carried out on a broad range of benchmark functions in global numerical optimization. The results show that LDE is very competitive, and both of the two main components have contributed to its overall performance. The scalability of LDE is also discussed by conducting experiments on some selected benchmark functions with dimensions from 30 to 200.     

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.     

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.     

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.     

Multi-population differential evolution with balanced ensemble of mutation strategies for large-scale global optimization

Differential evolution (DE) is a simple, yet very effective, population-based search technique. However, it is challenging to maintain a balance between exploration and exploitation behaviors of the DE algorithm. In this paper, we boost the population diversity while preserving simplicity by introducing a multi-population DE to solve large-scale global optimization problems. In the proposed algorithm, called mDE-bES, the population is divided into independent subgroups, each with different mutation and update strategies. A novel mutation strategy that uses information from either the best individual or a randomly selected one is used to produce quality solutions to balance exploration and exploitation. Selection of individuals for some of the tested mutation strategies utilizes fitness-based ranks of these individuals. Function evaluations are divided into epochs. At the end of each epoch, individuals between the subgroups are exchanged to facilitate information exchange at a slow pace. The performance of the algorithm is evaluated on a set of 19 large-scale continuous optimization problems. A comparative study is carried out with other state-of-the-art optimization techniques. The results show that mDE-bES has a competitive performance and scalability behavior compared to the contestant algorithms.     

加权变异策略动态差分进化算法

针对差分进化算法在解决高维优化问题时易早熟收敛、求解精度低和参数设置麻烦等问题,提出一种加权变异策略动态差分进化算法（WMDDE）。为了动态平衡全局搜索与局部搜索能力,跳出局部最优,将标准差分进化算法的变异策略DE/rand/1和DE/best/1进行加权组合,提出两种新的随机扰动加权变异算子。提出一种动态自适应调整缩放因子和交叉概率因子的策略,避免参数设置的麻烦,提高算法的稳定性。在11个Benchmark函数上的测试结果表明,新算法能有效避免早熟收敛,全局寻优能力强,且在高维时寻优速度、求解精度和稳定性均优于4种DE进化算法。     

Independent tracking and regulation adaptive control with forgetting factor

This paper considers a discrete-time adaptive control algorithm with a forgetting factor applicable to minimum phase plants. The tracking and regulation objectives are independently specified. The relevance of the eigenvalues of the gain matrix (F_k) used in the updating equation for the adaptive parameters ($\text{}\text{?}\text{gq(k)}$) is shown. It is proved that if the maximum eigenvalue of the inverse of the gain matrix F_k has an upper bound and a non-zero lower bound then the global convergence of the control algorithm is insured. The result of the design is a simple control scheme using a linear constant feedforward controller and a nonlinear feedback controller. The performance of the control structure in tracking and regulation are evaluated by simulations.     

An enhanced artificial bee colony algorithm with adaptive differential operators

Artificial bee colony algorithm (ABC) has been shown to be very effective to solve global optimization problems (GOPs). However, ABC performs well in exploration but relatively poorly in exploitation resulting in a slow convergence when it is used to handle complex GOPs. Differential evolution (DE) benefits from its differential operators, namely mutation operator and crossover operator, which could perturb multiple variables simultaneously and has shown a fast convergence speed. In order to improve ABC’s exploitation ability and accelerate its convergence, in this paper, we propose an enhanced ABC algorithm named ABCADE, which remedy the limitation of ABC by exploiting the advantage of differential operators. Particularly, in ABCADE, the employed bees employ differential operators to produce candidate solutions with an increasing probability, and the two important parameters (scale factor F and crossover rate CR) of differential operators are adaptively adjusted through Gaussian distribution. Moreover, to significantly differentiate the good solutions and bad solutions in a population, and put more effort in the exploitation around the good solutions, we design a new selection probability method for onlooker bees. To verify the performance of ABCADE, we compare ABCADE with other representative state-of-the-art ABC and DE algorithms, the comparison results on a set of 22 benchmark functions with various dimension sizes demonstrate that ABCADE obtains superior or comparable performance to other algorithms.     

Hybrid differential evolution with biogeography-based optimization algorithm for solution of economic emission load dispatch problems

This paper presents combination of differential evolution (DE) and biogeography-based optimization (BBO) algorithm to solve complex economic emission load dispatch (EELD) problems of thermal generators of power systems. Emission substances like NO_X, SO_X, CO_X, Power demand equality constraint and operating limit constraint are considered here. Differential evolution (DE) is one of the very fast and robust, accurate evolutionary algorithms for global optimization and solution of EELD problems. Biogeography-based optimization (BBO) is another new biogeography inspired algorithm. Biogeography deals with the geographical distribution of different biological species. This algorithm searches for the global optimum mainly through two steps: migration and mutation. In this paper combination of DE and BBO (DE/BBO) is proposed to accelerate the convergence speed of both the algorithm and to improve solution quality. To show the advantages of the proposed algorithm, it has been applied for solving multi-objective EELD problems in a 3-generator system with NO_X and SO_X emission, in a 6-generators system considering NO_X emission, in a 6-generator system addressing both valve-point loading and NO_X emission. The current proposal is found better in terms of quality of the compromising and individual solution obtained.     

Enhancing the search ability of differential evolution through orthogonal crossover

Differential evolution (DE) is a class of simple yet powerful evolutionary algorithms for global numerical optimization. Binomial crossover and exponential crossover are two commonly used crossover operators in current popular DE. It is noteworthy that these two operators can only generate a vertex of a hyper-rectangle defined by the mutant and target vectors. Therefore, the search ability of DE may be limited. Orthogonal crossover (OX) operators, which are based on orthogonal design, can make a systematic and rational search in a region defined by the parent solutions. In this paper, we have suggested a framework for using an OX in DE variants and proposed OXDE, a combination of DE/rand/1/bin and OX. Extensive experiments have been carried out to study OXDE and to demonstrate that our framework can also be used for improving the performance of other DE variants.     

A novel clustering-based differential evolution with 2 multi-parent crossovers for global optimization

Differential evolution (DE) is a simple and efficient global optimization algorithm. However, DE has been shown to have certain weaknesses, especially if the global optimum should be located using a limited number of function evaluations (NFEs). Hence hybridization with other methods is a research direction for the improvement of differential evolution. In this paper, a hybrid DE based on the one-step k-means clustering and 2 multi-parent crossovers, called clustering-based differential evolution with 2 multi-parent crossovers (2-MPCs-CDE) is proposed for the unconstrained global optimization problems. In 2-MPCs-CDE, k cluster centers and several new individuals generate two search spaces. These spaces are then searched in turn. This method utilizes the information of the population effectively and improves search efficiency. Hence it can enhance the performance of DE. A comprehensive set of 35 benchmark functions is employed for experimental verification. Experimental results indicate that 2-MPCs-CDE is effective and efficient. Compared with other state-of-the-art evolutionary algorithms, 2-MPCs-CDE performs better, or at least comparably, in terms of the solution accuracy and the convergence rate.     

A novel hybrid DE–random search approach for unit commitment problem

Differential evolution (DE) is a population-based stochastic search algorithm, whose simple yet powerful and straightforward features make it very attractive for numerical optimization. DE uses a rather greedy and less stochastic approach to problem-solving than other evolutionary algorithms. DE combines simple arithmetic operators with the classical operators of recombination, mutation and selection to evolve from a randomly generated starting population to a final solution. Although global exploration ability of DE algorithm is adequate, its local exploitation ability is feeble and convergence velocity is too low and it suffers from the problem of untime convergence for multimodal objective function, in which search process may be trapped in local optima and it loses its diversity. Also, it suffers from the stagnation problem, where the search process may infrequently stop proceeding toward the global optimum even though the population has not converged to a local optimum or any other point. To improve the exploitation ability and global performance of DE algorithm, a novel and hybrid version of DE algorithm is presented in the proposed research. This research paper presents a hybrid version of DE algorithm combined with random search for the solution of single-area unit commitment problem. The hybrid DE–random search algorithm is tested with IEEE benchmark systems consisting of 4, 10, 20 and 40 generating units. The effectiveness of proposed hybrid algorithm is compared with other well-known evolutionary, heuristics and meta-heuristics search algorithms, and by experimental analysis, it has been found that proposed algorithm yields global results for the solution of unit commitment problem.

     

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.     

A clustering method combining differential evolution with the K-means algorithm

The present paper considers the problem of partitioning a dataset into a known number of clusters using the sum of squared errors criterion (SSE). A new clustering method, called DE-KM, which combines differential evolution algorithm (DE) with the well known K-means procedure is described. In the method, the K-means algorithm is used to fine-tune each candidate solution obtained by mutation and crossover operators of DE. Additionally, a reordering procedure which allows the evolutionary algorithm to tackle the redundant representation problem is proposed. The performance of the DE-KM clustering method is compared to the performance of differential evolution, global K-means method, genetic K-means algorithm and two variants of the K-means algorithm. The experimental results show that if the number of clusters K is sufficiently large, DE-KM obtains solutions with lower SSE values than the other five algorithms.     

Differential Evolution for learning the classification method PROAFTN

This paper introduces a new learning technique for the multicriteria classification method PROAFTN. This new technique, called DEPRO, utilizes a Differential Evolution (DE) algorithm for learning and optimizing the output of the classification method PROAFTN. The limitation of the PROAFTN method is largely due to the set of parameters (e.g., intervals and weights) required to be obtained to perform the classification procedure. Therefore, a learning method is needed to induce and extract these parameters from data. DE is an efficient metaheuristic optimization algorithm based on a simple mathematical structure to mimic a complex process of evolution. Some of the advantages of DE over other global optimization methods are that it often converges faster and with more certainty than many other methods and it uses fewer control parameters. In this work, the DE algorithm is proposed to inductively obtain PROAFTN’s parameters from data to achieve a high classification accuracy. Based on results generated from 12 public datasets, DEPRO provides excellent results, outperforming the most common classification algorithms.     

An efficient hybrid technique for numerical optimization and applications

Real World Optimization Problems is one of the major concerns to show the potential and effectiveness of an optimization algorithm. In this context, a hybrid algorithm of two popular heuristics namely Differential Evolution (DE) and Particle Swarm Optimization (PSO) engaged on a ‘tri-population’ environment. 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 abbreviated as DPD as it uses DE–PSO–DE on a population. Two strategies namely Elitism (to retain the best obtained values so far) and Non-redundant search (to improve the solution quality) have been additionally employed in DPD cycle. Moreover, the robustness of the mutation strategies of DE have been well studied and suitable mutation strategies for both DEs (for DPD) are investigated over a set of existing 8 popular mutation strategies which results 64 variants of DPD. The top DPD is further tested through the test functions of CEC2006, CEC2010 and 5 Engineering Design Problems. Also it is used to solve CEC2011 Real World Optimization problems. An excellent efficiency of the recommended DPD is confirmed over the state-of-the-art algorithms.     

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.     

A neural differential evolution identification approach to nonlinear systems and modelling of shape memory alloy actuator

This paper proposes a hybrid modified differential evolution plus back‐propagation (MDE‐BP) algorithm to optimize the weights of the neural network model. In implementing the proposed training algorithm, the mutation phase of the differential evolution (DE) is modified by combining two mutation strategies rand/1 and best/1 to create trial vectors instead of only using one mutation operator or rand/1 or best/1 as the standard DE. The modification aims to balance the global exploration and local exploitation capacities of the algorithm in order to find potential global optimum solutions. Then the local searching ability of the back‐propagation (BP) algorithm is applied in that region so as to swiftly converge to the optimum solution. The performance and efficiency of the proposed method is tested by identifying some benchmark nonlinear systems and modeling the shape memory alloy actuator. The proposed training algorithm is compared with the other algorithms, such as the traditional DE and BP algorithm. As a result, the proposed method can improve the accuracy of the identification process.     

Differential evolution with Gaussian mutation for combined heat and power economic dispatch

This paper presents differential evolution with Gaussian mutation to solve the complex non-smooth non-convex combined heat and power economic dispatch (CHPED) problem. Valve-point loading and prohibited operating zones of conventional thermal generators are taken into account. Differential evolution (DE) is a simple yet powerful global optimization technique. It exploits the differences of randomly sampled pairs of objective vectors for its mutation process. This mutation process is not suitable for complex multimodal optimization. This paper proposes Gaussian mutation in DE which improves search efficiency and guarantees a high probability of obtaining the global optimum without significantly impairing the simplicity of the structure of DE. The effectiveness of the proposed method has been verified on five test problems and three test systems. The results of the proposed approach are compared with those obtained by other evolutionary methods. It is found that the proposed differential evolution with Gaussian mutation-based approach is able to provide better solution.     

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.