Control Parameters Of Differential Evolution Research Articles

Dimension improvements based adaptation of control parameters in Differential Evolution: A fitness-value-independent approach

In this paper, we proposed a novel fitness-value-independent Differential Evolution to tackle complex real-parameter single-objective optimization. There were three innovations in the algorithm: First, dimension improvements based adaptation schemes for control parameters F and CR were advanced in this paper. Different from those fitness value improvements based parameter control in the recent winner DE variants, our algorithm has fitness-value-independent characteristic, therefore, it can be applied in much wider optimization scenarios especially for those that the exact fitness values of the objectives are unavailable. Second, a combined parabolic–linear reduction scheme for population size is employed in the algorithm based on the observation that the slower reduction of population size at the earlier stage of the evolution usually helps to get better perception of the objectives while a linear reduction of population size in the later stage helps to get better exploitation. Third, an indicator is proposed to monitor the diversity of the population and a corresponding population enhancement technique is launched when the diversity is detected bad. The first two innovations have the Parameter adaptive characteristic of the DE algorithm while the third innovation refers to population enhancement technique, therefore, we name our algorithm “the PaDE-pet algorithm”. Then, the algorithm is validated under a larger test suite containing 88 benchmarks from CEC2013, CEC2014 and CEC2017 test suites for real-parameter single objective optimization, which may avoid over-fitting problem in comparison with employing a test suite containing a small number of benchmarks. The experiments support the superiority of the novel PaDE-pet algorithm in comparison with several recently proposed powerful DE variants.

Using spatial neighborhoods for parameter adaptation: An improved success history based differential evolution

Differential Evolution (DE) has been widely appraised as a simple yet robust population-based, non-convex optimization algorithm primarily designed for continuous optimization. Two important control parameters of DE are the scale factor F, which controls the amplitude of a perturbation step on the current solutions and the crossover rate Cr, which limits the mixing of components of the parent and the mutant individuals during recombination. We propose a very simple, yet effective, nearest spatial neighborhood-based modification to the adaptation process of the aforesaid parameters in the Success-History based adaptive DE (SHADE) algorithm. SHADE uses a historical archive of the successful F and Cr values to update these parameters and stands out as a very competitive DE variant of current interest. Our proposed modifications can be extended to any SHADE-based DE algorithm like L-SHADE (SHADE with linear population size reduction), jSO (L-SHADE with modified mutation) etc. The enhanced performance of the modified SHADE algorithm is showcased on the IEEE CEC (Congress on Evolutionary Computation) 2013, 2014, 2015, and 2017 benchmark suites by comparing against the DE-based winners of the corresponding competitions. Furthermore, the effectiveness of the proposed neighborhood-based parameter adaptation strategy is demonstrated by using the real-life problems from the IEEE CEC 2011 competition on testing evolutionary algorithms on real-world numerical optimization problems.

Deep-space trajectory optimizations using differential evolution with self-learning

This paper presents spacecraft trajectory optimizations for deep-space missions requiring multiple gravity-assists (MGA). The main algorithm is based on a self-adaptive/self-learning differential evolution (DE). In the process of improving the performance of DE for optimizing the MGA trajectory, the proposed algorithm alleviates the dependence on predetermined mutation strategy and control parameters in DE; as evolution progresses, the mutation strategy and the control parameters switch adaptively to more promising ones by reflecting experiences in previous evolution steps. Furthermore, the proposed algorithm is equipped with a re-initialization technique to directly mollify the issue of converging to a local optimum, which is often observed when optimizing the MGA trajectory. In order to demonstrate these favorable characteristics, the proposed algorithm is implemented to solve six well-known MGA trajectory optimization problems designed by the European space agency (ESA). Compared with the latest representative evolutionary algorithms, the proposed algorithm can successfully converge to the currently known best solutions of five MGA problems; our solutions to four of those MGA problems are better than currently known solutions. The proposed algorithm also performs well as a local/auxiliary search algorithm to improve the performance of other evolutionary algorithms. In addition to describing the algorithms and solutions characteristics, sensitivity analysis is presented to quantitatively investigate the search capability of finding the optimal solutions of MGA problems. The overall results show that our self-learning DE is competitively compared with other representative algorithms in terms of convergences to the global optimum, reliable search capability, and applicability to a variety of deep-space trajectory optimizations.

Using proposed optimization algorithm for solving inverse kinematics of human upper limb applying in rehabilitation robotic

The requirement to solve the problem of Inverse Kinetics (IK) plays a very important role in the robotics field in general, and especially in the field of rehabilitation robots, in particular. If the solutions of this problem are not suitable, it can cause undesirable damage to the patient when exercising. Normally, the problem of Inverse Kinematics in the robotics field, as well as the natural field, especially for redundant driven systems, often requires the application of a lot of techniques. The redundancy in Degree of Freedom (DoF), the nonlinearity of the system leads to solve inverse kinematics problem more challenge. In this study, we proposed to apply the self-adaptive control parameters in Differential Evolution with search space improvement (Pro-ISADE) to solve the problem for the human upper limb, which is a very typical redundancy model in nature. First of all, the angles of the joints were measured by a proposed Exoskeleton type Human Motion Capture System (E-HMCS) when the wearer performs some Activities of Daily Living (ADL) and athletic activities. The values of these measured angles joints then were put into the forward kinematics model to find the end effector trajectories. After having these orbits, they were re-fed into the proposed Pro-ISADE algorithm mentioned above to process the IK problem and obtain the predicted joints angular values. The experimental results showed that the predicted joints’ values closely follow the measured joints’ values. That demonstrates the ability to apply the Pro-ISADE algorithm to solve the problem of Inverse Kinetics of the human upper limb as well as the upper limb rehabilitation robot arm.

Adaptive differential evolution with a new joint parameter adaptation method

Differential evolution (DE) is a population-based metaheuristic algorithm that has been proved powerful in solving a wide range of real-parameter optimization tasks. However, the selection of the mutation strategy and control parameters in DE is problem dependent, and inappropriate specification of them will lead to poor performance of the algorithm such as slow convergence and early stagnation in a local optimum. This paper proposes a new method termed as Joint Adaptation of Parameters in DE (JAPDE). The key idea lies in dynamically updating the selection probabilities for a complete set of pairs of parameter generating functions based on feedback information acquired during the search by DE. Further, for mutation strategy adaptation, the Rank-Based Adaptation (RAM) method is utilized to facilitate the learning of multiple probability distributions, each of which corresponds to an interval of fitness ranks of individuals in the population. The coupling of RAM with JAPDE results in the new RAM-JAPDE algorithm that enables simultaneous adaptation of the selection probabilities for pairs of control parameters and mutation strategies in DE. The merit of RAM-JAPDE has been evaluated on the benchmark test suit proposed in CEC2014 in comparison to many well-known DE algorithms. The results of experiments demonstrate that the proposed RAM-JAPDE algorithm outperforms or is competitive to the other related DE variants that perform mutation strategy and control parameter adaptation, respectively.

An improved differential evolution algorithm using Archimedean spiral and neighborhood search based mutation approach for cluster analysis

This study proposes an improved Differential Evolution (DE) algorithm utilizing Archimedean Spiral, Mantegna Lévy Distribution, and Neighborhood Search (NS). The proposed algorithm is denoted as Adaptive Differential Evolution with Neighborhood Search (ADENS). The aim of the ADENS algorithm is to enhance the convergence speed and keeping a balance between exploration and exploitation of the DE algorithm. It uses a new mutation strategy to generate robust solutions by combining the Archimedean Spiral (AS) with the Mantegna Lévy flight. In order to enhance the efficiency of ADENS, a replacement method combines Lévy flight with neighborhood search to generate solutions that replace poorly performing ones. A self-adaptive strategy is applied to fine-tune one of the control parameters of DE and an initialization method is employed to initialize the algorithm. These strategies help the algorithm achieve good efficiency in terms of convergence speed and both local and global search. The performance is evaluated using twelve well-known standard data sets to show the algorithm’s superior performance, confirmed to be statistically significant. In order to test the proposed approach statistically, this paper applied Wilcoxon ranked sum test as well as Friedman test. Our results also shed light on the comparative performance of some recently published clustering heuristics. The results show that the algorithm is a robust algorithm and has a great superiority with respect to the employed algorithms. The proposed algorithm can be applied in different applications such as medical diagnosis and image segmentation according to the conducted experiments.

Opposition-based learning for self-adaptive control parameters in differential evolution for optimal mechanism design

In recent decades, new optimization algorithms have attracted much attention from researchers in both gradient- and evolution-based optimal methods. Many strategy techniques are employed to enhance the effectiveness of optimal methods. One of the newest techniques is opposition-based learning (OBL), which shows more power in enhancing various optimization methods. This research presents a new edition of the Differential Evolution (DE) algorithm in which the OBL technique is applied to investigate the opposite point of each candidate of self-adaptive control parameters. In comparison with conventional optimal methods, the proposed method is used to solve benchmark-test optimal problems and applied to real optimizations. Simulation results show the effectiveness and improvement compared with some reference methodologies in terms of the convergence speed and stability of optimal results. © 2019 The Japan Society of Mechanical Engineers

Finding High-Dimensional D-Optimal Designs for Logistic Models via Differential Evolution.

D-optimal designs are frequently used in controlled experiments to obtain the most accurate estimate of model parameters at minimal cost. Finding them can be a challenging task, especially when there are many factors in a nonlinear model. As the number of factors becomes large and interact with one another, there are many more variables to optimize and the D-optimal design problem becomes high-dimensional and non-separable. Consequently, premature convergence issues arise. Candidate solutions get trapped in local optima and the classical gradient-based optimization approaches to search for the D-optimal designs rarely succeed. We propose a specially designed version of differential evolution (DE) which is a representative gradient-free optimization approach to solve such high-dimensional optimization problems. The proposed specially designed DE uses a new novelty-based mutation strategy to explore the various regions in the search space. The exploration of the regions will be carried out differently from the previously explored regions and the diversity of the population can be preserved. The proposed novelty-based mutation strategy is collaborated with two common DE mutation strategies to balance exploration and exploitation at the early or medium stage of the evolution. Additionally, we adapt the control parameters of DE as the evolution proceeds. Using logistic models with several factors on various design spaces as examples, our simulation results show our algorithm can find D-optimal designs efficiently and the algorithm outperforms its competitors. As an application, we apply our algorithm and re-design a 10-factor car refueling experiment with discrete and continuous factors and selected pairwise interactions. Our proposed algorithm was able to consistently outperform the other algorithms and find a more efficient D-optimal design for the problem.

Adaptation of operators and continuous control parameters in differential evolution for constrained optimization

A large number of differential evolution algorithms has been proposed in recent years, many of which have been used to solve mainly unconstrained problems. However, similar to other evolutionary algorithms, their performances are highly dependent on their search operators and control parameters. Although many investigations have been conducted to ensure appropriate choices of these operators and parameters, the task is recognized as tedious. In this research, a differential evolution algorithm, which includes a new mechanism for automatically selecting the best combinations of parameters (amplification factor, crossover rate, and population size) as well as search operators, is developed. Instead of choosing discrete values for the amplification factor and crossover rate from a given set of values, this study adaptively selects them from some given continuous ranges and, furthermore, proposes a new methodology for handling constraints. The performance of the algorithm is assessed using a well-known set of constrained problems, with the experimental results demonstrating that it is superior to state-of-the-art algorithms.

Understanding differential evolution: A Poisson law derived from population interaction network

Differential evolution (DE) is one of evolutionary algorithms to effectively handle optimization problems. We propose a population interaction network (PIN) to investigate the relationship constituted by populations. The cumulative distribution function (CDF) of degree in PIN is analyzed by five fitting models on 12 benchmark functions. The goodness of fit is used to measure the fitting results. The experimental results demonstrate the CDF meets cumulative Poisson distribution. Besides, the number of nodes in PIN and the rate parameter λ in the fitted Poisson distribution are further studied using different control parameters of DE, which exhibits the effect and characteristic of the population interaction.

An Enhanced Jaya Algorithm with a Two Group Adaption

This paper proposes a novel performance enhanced Jaya algorithm with a two group adaption (E-Jaya). Two improvements are presented in E-Jaya. First, instead of using the best and the worst values in Jaya algorithm, E-Jaya separates all candidates into two groups: the better and the worse groups based on their fitness values, then the mean of the better group and the mean of the worse group are used. Second, in order to add non algorithm-specific parameters in E-Jaya, a novel adaptive method of dividing the two groups has been developed. Finally, twelve benchmark functions with different dimensionality, such as 40, 60, and 100, were evaluated using the proposed E-Jaya algorithm. The results show that E-Jaya significantly outperformed Jaya algorithm in terms of the solution accuracy. Additionally, E-Jaya was also compared with a differential evolution (DE), a self-adapting control parameters in differential evolution (jDE), a firefly algorithm (FA), and a standard particle swarm optimization 2011 (SPSO2011) algorithm. E-Jaya algorithm outperforms all the algorithms.

Opposition-Based Adaptive Fireworks Algorithm

A fireworks algorithm (FWA) is a recent swarm intelligence algorithm that is inspired by observing fireworks explosions. An adaptive fireworks algorithm (AFWA) proposes additional adaptive amplitudes to improve the performance of the enhanced fireworks algorithm (EFWA). The purpose of this paper is to add opposition-based learning (OBL) to AFWA with the goal of further boosting performance and achieving global optimization. Twelve benchmark functions are tested in use of an opposition-based adaptive fireworks algorithm (OAFWA). The final results conclude that OAFWA significantly outperformed EFWA and AFWA in terms of solution accuracy. Additionally, OAFWA was compared with a bat algorithm (BA), differential evolution (DE), self-adapting control parameters in differential evolution (jDE), a firefly algorithm (FA), and a standard particle swarm optimization 2011 (SPSO2011) algorithm. The research results indicate that OAFWA ranks the highest of the six algorithms for both solution accuracy and runtime cost.

Design of self-adaptive and equilibrium differential evolution optimized radial basis function neural network classifier for imputed database

The occurrence of missing values is not uncommon in real life databases like industrial, medical, and life science. The imputation of these values has been realized through the mean/mode of known values (for a quantitative/qualitative attribute) or nearest neighbors. Mean based imputation considerably underestimates the population variance and tends to weaken the attribute relationships. Similarly, the nearest neighbor approach uses only information of the nearest neighbors and leaving other observations aside. Hence to overcome the shortcomings of these methods, we have introduced a method known as medoid based imputation to impute missing values. Further, to achieve better performance, we have devised a novel classifier for imputed datasets, by using the self-adaptive control parameters of differential evolution (DE) with equilibrium of exploitation and exploration optimized radial basis function neural networks (RBFNs). By newly associating a weight parameter with target vector during mutation, we maintain equilibrium on the exploration and exploitation mechanism of DE. The self-adaptive equilibrium DE (SAEDE) is used to explore and exploit the suitable kernel parameters of RBFNs along with bias and then used for classifying unknown samples. The performance of the proposed classifier named as SAEDE-RBFN has been extensively evaluated on seven datasets retrieved from University of California, Irvine (UCI) and KEEL machine learning repositories after imputation by mean, nearest neighbor, and proposed method. The average performance of classifiers has been listed based on the imputation by K-nearest neighbor (Knn = 1, Knn = 3, Knn = 5, and Knn = 7), mean, and medoid, respectively. Outcome of the experimental study shows that the performance of SAEDE-RBFN on medoid based imputed dataset is relatively better than DE-RBFN.

Hybridization of Adaptive Differential Evolution with an Expensive Local Search Method

Differential evolution (DE) is an effective and efficient heuristic for global optimization problems. However, it faces difficulty in exploiting the local region around the approximate solution. To handle this issue, local search (LS) techniques could be hybridized with DE to improve its local search capability. In this work, we hybridize an updated version of DE, adaptive differential evolution with optional external archive (JADE) with an expensive LS method, Broydon-Fletcher-Goldfarb-Shano (BFGS) for solving continuous unconstrained global optimization problems. The new hybrid algorithm is denoted by DEELS. To validate the performance of DEELS, we carried out extensive experiments on well known test problems suits, CEC2005 and CEC2010. The experimental results, in terms of function error values, success rate, and some other statistics, are compared with some of the state-of-the-art algorithms, self-adaptive control parameters in differential evolution (jDE), sequential DE enhanced by neighborhood search for large-scale global optimization (SDENS), and differential ant-stigmergy algorithm (DASA). These comparisons reveal that DEELS outperforms jDE and SDENS except DASA on the majority of test instances.

Diversity Controlled Self Adaptive Differential Evolution based design of non-fragile multivariable PI controller

This paper proposes the design of non-fragile multivariable PI controller for industrial Multi Input Multi Output (MIMO) processes using Diversity Controlled Self-Adaptive Differential Evolution with Local Search (DCSaDE-LS) algorithm. The proposed non-fragile multivariable PI controller design is formulated as a single objective optimization problem with the objective of robustness subject to non-fragileness constraint. Monte Carlo Simulation is used to evaluate the non-fragileness constraint against stochastic uncertainty of controller parameters. Wood and Berry Distillation process and a real time Level-Temperature reactor process model with simple de-coupler are considered as a test systems. For constraint handling, Constraint Domination Principle (CDP) and Penalty Parameter Less (PPL) methods are incorporated in DCSaDE-LS, seperately. Also, the performance of DCSaDE-LS is compared with the recent adaptive DE algorithms namely Ensemble of mutation strategies and control parameters in Differential Evolution (EPSDE), Composite Differential Evolution (CoDE) and SaDE. The robustness and non-fragileness of the controller is analyzed under +40% and −40% variations in nominal process and controller parameters. The results demonstrated that the DCSaDE-LS designed non-fragile multivariable PI controller has good time domain performance, better robustness and non-fragileness than the reported multivariable PI controllers namely Maghade-PI, IMC-PI and NDT-PI.

Performance Enhancement of Differential Evolution by Incorporating Lévy Flight and Chaotic Sequence for the Cases of Satellite Images

Differential Evolution (DE) is a simple but powerful evolutionary algorithm. Crossover Rate (CR) and Mutation Factor (F) are the most important control parameters in DE. Mutation factor controls the diversification. In traditional DE algorithm CR and F are kept constant. In this paper, the values of CR and F are modified to enhance the capability of traditional DE algorithm. In the first modified algorithm chaotic sequence is used to perform this modification. In the next modified algorithm Lévy Flight with chaotic step size is used for such enhancement. In the second modified DE, population diversity has been used to build population in every generation. As a result the algorithm does not converge prematurely. Both modified algorithms have been applied to optimize parameters of the parameterized contrast stretching function. The algorithms are tested for satellite image contrast enhancement and the results are compared, which show that DE via chaotic Lévy and population diversity information outperforms the traditional and chaotic DE in the image enhancement domain.

Differential evolution improved with self-adaptive control parameters based on simulated annealing

Nowadays, differential evolution (DE) has attracted more and more attention as an effective approach for solving numerical optimization problems. However, the fact that users have to set the control parameters of DE according to every single different problem makes the adjustment of control parameters a very time-consuming work. To solve the problem, this paper presents an enhanced self-adaptive differential evolution (ESADE) for global numerical optimization over continuous space. In this ESADE algorithm, different control parameters have been used to make mutation and crossover. Here is the detailed process: Firstly, it initializes two groups of population. Secondly, it generates a set of control parameters for one of the two populations and then further creates another new series of control parameters for the other population through mutating the initial control parameters. Thirdly, once the control parameters are generated, the two populations are mutated and crossed to produce two groups of trial vectors. Finally, the target vectors are selected from the two groups of trial vectors by selecting operation. In order to enhance its global search capabilities, simulated annealing (SA) are involved in the selecting operation and the control parameters with better performance are chosen as the initial control parameters of the next generation. By employing a set of 17 benchmark functions from previous literature, this study carried out extensive computational simulations and comparisons and the computational results showed that the ESADE algorithm generally performed better than the state-of-the-art differential evolution variants and PSO. Besides, the influences of initialized ambient temperature and simulated annealing on the performance of ESADE have also been tested. For the purpose of testing the application of ESADE in solving real-world problems, ESADE was applied to identify the parameters of proton exchange membrane fuel cell model. The results showed that ESADE was equal with other state-of-the-art differential evolution variants on performance.

A new self-adaptation scheme for differential evolution

The performance of Differential Evolution (DE) largely depends on the choice of trial vector generation strategy and the values of its control parameters. In the past years, quite a few DE variants have been developed to adaptively adjust the strategy and control parameters during the search process. However, these variants may not perform satisfactorily when coping with computationally expensive problems (CEPs) for which a satisfying solution needs to be obtained with very limited fitness evaluations (FEs). In this paper, we demonstrate that not only can surrogate models be used to approximate the fitness function, they can also provide a good alternative method to adapt the strategy and control parameters of DE, and thus propose a framework called DE with Surrogate-assisted Self-Adaptation (DESSA). DESSA generates multiple trial vectors using different trial vector generation strategies and parameter settings, and then employs a surrogate model to identify the potentially best trial vector to undergo real fitness evaluation. As each trial vector corresponds to a unique combination of strategy and parameter setting, the surrogate model acts like a strategy/parameter setting selector that aims to identify the most suitable strategy and parameter setting for each target vector. Since DESSA can be easily combined with different DE variants, three concrete DE variants, namely DESSA-CoDE, DESSA-SaDE, and DESSA-CoDE*, are proposed. Comprehensive empirical studies demonstrate that DESSA can lead to superior performance over the compared adaptive DE variants. More importantly, it is shown that DESSA has the potential of accommodating more search strategies, which may lead to novel DE variants with even more competitive performance.

Enhanced lipase recovery through RSM integrated differential evolutionary approach from the fermented biomass

The aim of this work was to apply a modeling integrated optimisation approach for a complex, highly nonlinear system for an extracellular lipase extraction process. The model was developed using mutation, crossover and selection variables of Differential Evolution (DE) based on central composite design of Response Surface Methodology. The experimentally validated model was optimized by DE, a robust evolutionary optimization tool. A maximum lipase activity of 134.13 U/gds (more than 36.28 U/gds compared to one variable at a time approach) was observed with the DE-stated optimum values of 25.01% dimethyl sulfoxide concentration, 40 mM buffer, 128.52 min soaking time and 35oC with the DE control parameters, namely number of population, generations, crossover operator and scaling factor as 20, 50, 0.5 and 0.25, respectively. The use of DE approach improved the optimization capability and decision speed, resulting in an improved yield of 36.28 U/gds compared to the one variable at a time approach for the extracellular lipase activity under the non-optimized conditions. The developed mathematical model and optimization were generic in nature, which seemed to be useful for the scale-up studies of maximum recovery of lipase from the fermented biomass.

Controller parameters tuning of differential evolution algorithm and its application to load frequency control of multi-source power system

This paper presents controller parameters tuning of Differential Evolution (DE) algorithm and its application to Load Frequency Control (LFC) of a multi-source power system having different sources of power generation like thermal, hydro and gas power plants. Initially, a single area multi-source power system with integral controllers for each unit is considered and DE technique is applied to obtain the controller parameters. Various mutation strategies of DE are compared and the control parameters of DE for best obtained strategy are tunned by executing multiple runs of algorithm for each parameter variation. The study is further extended to a multi-area multi-source power system and a HVDC link is also considered in parallel with existing AC tie line for the interconnection of two areas. The parameters of Integral (I), Proportional Integral (PI) and Proportional Integral Derivative (PID) are optimized employing tunned DE algorithm. The superiority of the proposed approach has been shown by comparing the results with recently published optimal output feedback controller for the same power systems. The comparison is done using various performance measures like overshoot, settling time and standard error criteria of frequency and tie-line power deviation following a step load perturbation (SLP). It is noticed that, the dynamic performance of proposed controller is better than optimal output feedback controller. Furthermore, it is also seen that the proposed system is robust and is not affected by change in the loading condition, system parameters and size of SLP.