Classical Differential Evolution Algorithm Research Articles

An efficient surrogate-assisted differential evolution algorithm for turbomachinery cascades optimization with more than 100 variables

The CFD-based design optimization of turbomachinery cascades is a typical high-dimensional expensive black-box (HEB) problem. Specifically, to fully consider the interactions between vanes and stages and thus achieving desirable solutions, the cascades of two or more stages have to be optimized simultaneously. The number of variables can reach to more than 100 while the affordable CFD simulations within a design cycle can be only a few hundred or thousand. Hence, how to achieve the optimal design of turbomachinery cascades on budget is very challenging. To solve this problem, a generalized surrogate-assisted differential evolution (DE) algorithm is proposed, which is labeled as GSDE. In GSDE, radial basis function is incorporated into the population regeneration process, which helps to enhance the local exploitation in each iteration and thus achieving a good balance in between global and local search. By validating on benchmark functions ranging from 50 to 100 dimensions, the proposed GSDE algorithm is observed to obtain the true optimal solution with less than 1000 function calls and have far better convergence rate than other state-of-the-art algorithms. Furthermore, GSDE is then tested on the cascades design optimization of a 3.5-stage compressor with 126 variables. Within 1500 CFD evaluations, the total-to-total efficiency is increased by 1.104%. In contrast, classic DE algorithm achieves a similar efficiency gain with nearly 20,000 CFD evaluations. In other words, the computational cost of GSDE amount to only 1/13 of frequently-used evolutionary algorithms. Therefore, the efficiency of the proposed GSDE algorithm for solving high-dimensional turbomachinery optimization problems is well demonstrated.

Improved Differential Evolution Algorithm for Slab Allocation and Hot-Rolling Scheduling Integration Problem

To reduce logistics scheduling costs and energy consumption, this paper studies the slab allocation and hot-rolling scheduling integrated optimization problem that arises in practical iron and steel enterprises. In this problem, slabs are first allocated to orders and then sent to heating furnaces for heating; then, they are sent to a hot-rolling mill for rolling. A 0–1 integer programming model is established to minimize the attribute difference in the allocation cost between slabs and orders, the switching cost of hot-rolling processing, and waiting times after slabs reach rolling mills. Given the problem’s characteristics, an improved differential evolution algorithm using a real-number coding method is designed to solve it. Three different heuristic algorithms are proposed to improve the quality of solutions in the initial population. Multiple parent individuals participate in the mutation operation, which increases the population diversity and prevents the algorithm from falling into the local optimum prematurely. Experiments on 14 sets of real production data from a large domestic iron and steel plant show that our improved differential evolution algorithm generates significantly better solutions in a reasonable amount of time compared with CPLEX, the simulated artificial method, and the classical differential evolution algorithm, and it can be used by practitioners.

An enhanced distributed differential evolution algorithm for portfolio optimization problems

The population structure of differential evolution (DE) algorithm cannot maintain the diversity of the population to the greatest extent and help the population avoid to fall into the local optima in time. In this paper, a co-evolutionary multi-swarm adaptive differential evolution algorithm, namely ECMADE is proposed to solve the premature convergence and search stagnation. First of all, in terms of population structure, based on the parallel distributed framework, ECMADE randomly and evenly divides the population into exploration subpopulation, development subpopulation, and auxiliary subpopulation, and introduces an adaptive information exchange mechanism so that subpopulations can escape local optima in time. Then, a multi-operator parallel search strategy is proposed to keep population diversity and meet the optimization needs of different problems. Finally, an adaptive adjustment mechanism of control parameters is developed, through recent elite parameter archive and weight distribution to fully mine successful parameter information, and generate control parameters with a high success rate for the current evolutionary stage. In order to prove the effectiveness of the ECMADE, 10 test functions and portfolio optimization problem are selected in here. The experiment results show that the ECMADE can effectively solve these test functions, the accuracy and efficiency is superior to those of two classical DE algorithms. The actual application results show that the ECMADE can significantly improve the ability of portfolio to resist extreme losses, which proves the effectiveness and feasibility of the ECMADE once again. The ECMADE has better optimization performance by comparing with some well-known algorithms in term of the solution quality, robustness and space distribution. It provides a new algorithm for solving complex optimization problems.

Segmentation on remote sensing imagery for atmospheric air pollution using divergent differential evolution algorithm.

Air pollution is a global issue causing major health hazards. By proper monitoring of air quality, actions can be taken to control air pollution. Satellite remote sensing is an effective way to monitor global atmosphere. Various sensors and instruments fitted to satellites and airplanes are used to obtain the radar images. These images are quite complex with various wavelength differentiated by very close color differences. Clustering of such images based on its wavelengths can provide the much-needed relief in better understanding of these complex images. Such task related to image segmentation is a universal optimization issue that can be resolved with evolutionary techniques. Differential Evolution (DE) is a fairly fast and operative parallel search algorithm. Though classical DE algorithm is popular, there is a need for varying the mutation strategy for enhancing the performance for varied applications. Several alternatives of classical DE are considered by altering the trial vector and control parameter. In this work, a new alteration of DE technique labeled as DiDE (Divergent Differential Evolution Algorithm) is anticipated. The outcomes of this algorithm were tested and verified with the traditional DE techniques using fifteen benchmark functions. The new variant DiDE exhibited much superior outcomes compared to traditional approaches. The novel approach was then applied on remote sensing imagery collected form TEMIS, a web based service for atmospheric satellite images and the image was segmented. Fuzzy Tsallis entropy method of multi-level thresholding technique is applied over DiDE to develop image segmentation. The outcomes obtained were related with the segmented results using traditional DE and the outcome attained was found to be improved profoundly. Experimental results illustrate that by acquainting DiDE in multilevel thresholding, the computational delay was greatly condensed and the image quality was significantly improved.

Optimal Allocation of Distributed Generators in Active Distribution Networks Using a New Oppositional Hybrid Sine Cosine Muted Differential Evolution Algorithm

The research proposes a new oppositional sine cosine muted differential evolution algorithm (O-SCMDEA) for the optimal allocation of distributed generators (OADG) in active power distribution networks. The suggested approach employs a hybridization of the classic differential evolution algorithm and the sine cosine algorithm in order to incorporate the exploitation and exploration capabilities of the differential evolution algorithm and the sine cosine algorithm, respectively. Further, the convergence speed of the proposed algorithm is accelerated through the judicious application of opposition-based learning. The OADG is solved by considering three separate mono-objectives (real power loss minimization, voltage deviation improvement and maximization of the voltage stability index) and a multi-objective framework combining the above three. OADG is also addressed for DGs operating at the unity power factor and lagging power factor while meeting the pragmatic operational requirements of the system. The suggested algorithm for multiple DG allocation is evaluated using a small test distribution network (33 bus) and two bigger test distribution networks (118 bus and 136 bus). The results are also compared to recent state-of-the-art metaheuristic techniques, demonstrating the superiority of the proposed method for solving OADG, particularly for large-scale distribution networks. Statistical analysis is also performed to showcase the genuineness and robustness of the obtained results. A post hoc analysis using Friedman–ANOVA and Wilcoxon signed-rank tests reveals that the results are of statistical significance.

A New Study on Optimization of Four-Bar Mechanisms Based on a Hybrid-Combined Differential Evolution and Jaya Algorithm

In mechanism design with symmetrical or asymmetrical motions, obtaining high precision of the input path given by working requirements of mechanisms can be a challenge for dimensional optimization. This study proposed a novel hybrid-combined differential evolution (DE) and Jaya algorithm for the dimensional synthesis of four-bar mechanisms with symmetrical motions, called HCDJ. The suggested algorithm uses modified initialization, a hybrid-combined mutation between the classical DE and Jaya algorithm, and the elitist selection. The modified initialization allows generating initial individuals, which are satisfied with Grashof’s condition and consequential constraints. In the hybrid-combined mutation, three differential groups of mutations are combined. DE/best/1 and DE/best/2, DE/current to best/1 and Jaya operator, and DE/rand/1, and DE/rand/2 belong to the first, second, and third groups, respectively. In the second group, DE/current to best/1 is hybrid with the Jaya operator. Additionally, the elitist selection is also applied in HCDJ to find the best solutions for the next generation. To validate the feasibility of HCDJ, the numerical examples of the symmetrical motion of four-bar mechanisms are investigated. From the results, the proposed algorithm can provide accurate optimal solutions that are better than the original DE and Jaya methods, and its solutions are even better than those of many other algorithms that are available in the literature.

An analysis of why cuckoo search does not bring any novel ideas to optimization

It has been more than 10 years since the first version of cuckoo search was proposed by Yang and Deb and published in the proceedings of the World Congress on Nature & Biologically Inspired Computing, in 2009. The two main articles on cuckoo search have now been cited almost 8700 times (according to Google scholar), there are books and chapters published about this algorithm, and a special issue has been organized to celebrate its first decade of existence. Given the popularity of the algorithm and its widespread use, it is quite surprising that no one has ever formally investigated its obvious resemblance to evolutionary algorithms. In this article, we conduct a rigorous analysis of cuckoo search in which we identify the concepts used in the algorithm and show that these are the exact same concepts proposed in the (μ+λ)–evolution strategy, a well-known evolutionary algorithm introduced originally in 1981, and in the classic differential evolution algorithm introduced in 1997. We analyze the “cuckoos’ parasitic behavior” metaphor that inspired the algorithm according to three criteria—usefulness, novelty and sound motivation—that allow to clarify whether the use of the metaphor is justified or not. The result is that cuckoo search does not comply with any of these criteria. Surprisingly, we found that the algorithm the authors proposed for cuckoo search does not match the publicly available implementation they provided; moreover, neither of them really follows the metaphor of the cuckoos that inspired the algorithm.

Multimodal multi-objective differential evolution algorithm based on spectral clustering

In recent years, in the face of the same problem in industrial production and life, decision-makers often hope to have a variety of different solutions to deal with. In other words, we hope to locate more different Pareto solutions under the condition of finding Pareto front. However, there are few researches in this field. For this reason, we propose a multimodal multi-objective differential evolution algorithm based on spectral clustering (SC-MMODE), which mainly uses some mechanisms to divide the solutions in the decision space into several mutually independent subpopulations. First, SC-MMODE uses a spectral clustering algorithm to control the decision space and form multiple sub-populations with good neighbourhood relations. Secondly, a special crowding distance mechanism is used to balance the distribution of solutions in the decision space and objective space. In addition, the classical differential evolution algorithm can effectively prevent premature convergence. Then, in 17 test problems, the SC-MMODE algorithm and some new multimode multi-objective algorithms are tested simultaneously. Finally, through the analysis of experimental data, the SC-MMODE algorithm can find more Pareto optimal sets in the decision space, so it can effectively solve such problems.

Dual mutations collaboration mechanism with elites guiding and inferiors eliminating techniques for differential evolution

Differential evolution (DE) is a powerful evolutionary algorithm for global optimization problems. Generally, appropriate mutation strategies and proper equilibrium between global exploration and local exploitation are significant to the performance of DE. From this consideration, in this paper, we present a novel DE variant, abbreviated to DMIE-DE, to further enhance the optimization capacity of DE by developing a dual mutations collaboration mechanism with elites guiding and inferiors eliminating techniques. More specifically, an explorative mutation strategy DE/current-to-embest with an elite individual serving as part of the difference vector and an exploitative mutation strategy DE/ebest-to-rand with selecting an elite individual as the base vector are employed simultaneously to achieve the balance between local and global performance of the whole population instead of only one mutation strategy in classical DE algorithm. The control parameters F and CR for above mutation strategies are updated adaptively to supplement the optimization ability of DMIE-DE based on a rational probability distribution model and the successful experience from the previous iterations. Moreover, an inferior solutions eliminating technique is embedded to enhance the convergence speed and compensate cost of the fitness evaluation times during the evaluation process. To evaluate the performance of DMIE-DE, experiments are conducted by comparing with five state-of-the-art DE variants on solving 29 test functions in CEC2017 benchmark set. The experimental results indicate that the performance of DMIE-DE is significantly better than, or at least comparable to the considered DE variants.

Design Scalability Study of the Γ-Shaped Piezoelectric Harvester Based on Generalized Classical Ritz Method and Optimization

This paper studies the design scalability of a Γ-shaped piezoelectric energy harvester (ΓEH) using the generalized classical Ritz method (GCRM) and differential evolution algorithm. The generalized classical Ritz method (GCRM) is the advanced version of the classical Ritz method (CRM) that can handle a multibody system by assembling its equations of motion interconnected by the constraint equations. In this study, the GCRM is extended for analysis of the piezoelectric energy harvesters with material and/or orientation discontinuity between members. The electromechanical equations of motion are derived for the PE harvester using GCRM, and the accuracy of the numerical simulation is experimentally validated by comparing frequency response functions for voltage and power output. Then the GCRM is used in the power maximization design study that considers four different total masses—15 g, 30 g, 45 g, 60 g—to understand design scalability. The optimized ΓEH has the maximum normalized power density of 23.1 × 103 kg·s·m−3 which is the highest among the reviewed PE harvesters. We discuss how the design parameters need to be determined at different harvester scales.

Q-Learning-based parameter control in differential evolution for structural optimization

The operations of metaheuristic optimization algorithms depend heavily on the setting of control parameters. Therefore the addition of adaptive control parameter has been widely studied and shown to enhance the problem flexibility and overall performance of the algorithms. This paper proposes Q-learning Differential Evolution (qlDE) algorithm, an adaptive control parameter Differential Evolution(DE) algorithm, for structural optimization. The reinforcement learning Q-learning model is integrated into DE as an adaptive parameter controller, adaptively adjusting control parameters of the algorithm at each search iteration in order to optimally regulate its behavior for different search domains. Moreover, by automatically controlling the balance of exploration and exploitation at different stages of the process, the performance of the optimizer will also be enhanced. To verify the effectiveness and robustness of the proposed qlDE algorithm in comparison with the classical DE and several other algorithms in the literature, five benchmark examples of truss structural weight minimizations will be performed in this study.

A systematic identification approach for biaxial piezoelectric stage with coupled Duhem-type hysteresis

PurposeThe problem of parameter identification for biaxial piezoelectric stages is still a challenging task because of the existing hysteresis, dynamics and cross-axis coupling. This study aims to find an accurate and systematic approach to tackle this problem.Design/methodology/approachFirst, a dual-input and dual-output (DIDO) model with Duhem-type hysteresis is proposed to depict the dynamic behavior of the biaxial piezoelectric stage. Then, a systematic identification approach based on a modified differential evolution (DE) algorithm is proposed to identify the unknown parameters of the Duhem-type DIDO model for a biaxial piezostage. The randomness and parallelism of the modified DE algorithm guarantee its high efficiency.FindingsThe experimental results show that the characteristics of the biaxial piezoelectric stage can be identified with adequate accuracy based on the input–output data, and the peak-valley errors account for 2.8% of the full range in the X direction and 1.5% in the Y direction. The attained results validated the correctness and effectiveness of the presented identification method.Originality/valueThe classical DE algorithm has many adjustment parameters, which increases the inconvenience and difficulty of using in practice. The parameter identification of Duhem-type DIDO piezoelectric model is rarely studied in detail and its successful application based on DE algorithm on a biaxial piezostage is hitherto unexplored. To close this gap, this work proposed a modified DE-based systematic identification approach. It not only can identify this complicated model with more parameters, but also has little tuning parameters and thus is easy to use.

Optimization to optimization (OtoO): optimize monarchy butterfly method with stochastics multi-parameter divergence method for benchmark functions and load frequency control

Optimization to optimization (OtoO) approach is proposed in this study. It aims to increase an optimization algorithm performance. OtoO approach has two types of optimization methods. First is essential algorithm, which is used for solution of the basic problem. Second is auxiliary algorithm that adjusted the parameters of the essential algorithm. In this study, the monarchy butterfly optimization (MBO) method and stochastic multi-parameter divergence optimization (SMDO) method were defined as essential algorithm and auxiliary algorithm, respectively. Constant parameters of the MBO method that affect performance (Keep, Max. Step Size, period and BAR) are primarily optimized on benchmark functions with the SMDO algorithm, and results are compared with each other and classical MBO, ABC (Artificial Bee Colony), ACO (Ant Colony), BBO (Biogeography-based), SGA (Simple Genetic) and DE (Differential Evolution) algorithms. In addition, OtoO approach is also tried via composite benchmark functions. In addition, PI and PID controllers were designed for the load frequency control of a hybrid power system. Results are compared with the FA (Firefly Algorithm) and GA (Genetic Algorithm) results. Results demonstrate that the performance of algorithms can be increased without disrupting the basic philosophy of algorithms and hybridizing algorithms with the proposed OtoO approach via benchmark functions and engineering problems.

An adaptive mutation strategy for differential evolution algorithm based on particle swarm optimization

Differential evolution (DE) algorithm is a very effective algorithm used for solving wide range of optimization problems. However, the performance of DE is dependent on the control parameters and to choose the right parameter value and tuning of these parameters is a challenging task. Therefore, a novel variant of differential evolution algorithm based on particle swarm optimization (DEPSO) is proposed to improve the overall performance of Differential evolution algorithm. In our proposed approach, we are using DE mutation strategy during the initial phase of evolution and therefore enlarge its search space possibly to the extent that helps in finding more encouraging results and thus avoid premature convergence. During the subsequent phase of evolution process, this value of sigmoid function reduces with the increase of number of iterations. In this scenario, there is a greater probability of operating PSO mutation strategy and thus this sigmoid function helps in improving the precision and convergence speed. The Performance of our proposed algorithm is tested with 10 benchmark test functions on 50 and 25 dimensions set, also tested with 11 test functions on 30- and 100-dimension test functions. We have also tested our proposed algorithm with 8 test functions on high dimension set as 500- and 1000-dimensions. The performance comparison shows that our proposed variant is giving significant improvement in convergence speed and thus avoiding premature convergence. Average performance of DEPSO is better than classical DE, PSO and other algorithms in comparison.

Enhanced Differential Evolution Algorithm with Local Search Based on Hadamard Matrix.

Differential evolution (DE) is a robust algorithm of global optimization which has been used for solving many of the real-world applications since it was proposed. However, binomial crossover does not allow for a sufficiently effective search in local space. DE's local search performance is therefore relatively poor. In particular, DE is applied to solve the complex optimization problem. In this case, inefficiency in local research seriously limits its overall performance. To overcome this disadvantage, this paper introduces a new local search scheme based on Hadamard matrix (HLS). The HLS improves the probability of finding the optimal solution through producing multiple offspring in the local space built by the target individual and its descendants. The HLS has been implemented in four classical DE algorithms and jDE, a variant of DE. The experiments are carried out on a set of widely used benchmark functions. For 20 benchmark problems, the four DE schemes using HLS have better results than the corresponding DE schemes, accounting for 80%, 75%, 65%, and 65% respectively. Also, the performance of jDE with HLS is better than that of jDE on 50% test problems. The experimental results and statistical analysis have revealed that HLS could effectively improve the overall performance of DE and jDE.

Hybrid Harmony Search Differential Evolution Algorithm

Differential evolution (DE) algorithm has some excellent attributes including strong exploration capability. However, it cannot balance the exploitation with exploration ability in the search process. To enhance the performance of the DE algorithm, this paper proposes a new algorithm named hybrid harmony differential evolution algorithm (HHSDE). The key features of HHSDE algorithm are as follows. First, a new mutation operation is developed for improving the efficiency of mutation, in which the New Harmony generation mechanics of the harmony algorithm (HS) is employed. Second, the harmony memory size is updated with the iteration. Third, a self-adaptive parameter adjustment strategy is presented to control scaling factor. Fourth, a new evaluation method is proposed to effectively assess the algorithm convergence performance. Two classical DE algorithms, HS algorithm, improvement Differential evolution algorithm(ISDE) and Hybrid Artificial Bee Colony algorithm with Differential Evolution(HABCDE) have been tested against HHSDE based on 25 benchmark functions of CEC2005 and the results reveal that the proposed algorithm is better than the other algorithms under consideration.

Solving Fuzzy Job-Shop Scheduling Problem Using DE Algorithm Improved by a Selection Mechanism

The emergence of fuzzy sets makes job-shop scheduling problem (JSSP) become better aligned with the reality. This article addresses the JSSP with fuzzy execution time and fuzzy completion time (FJSSP). We choose the classic differential evolution (DE) algorithm as the basic optimization framework. The advantage of the DE algorithm is that it uses a special evolutionary strategy of difference vector sets to carry out mutation operation. However, DE is not very effective in solving some instances of FJSSP. Therefore, we propose a novel selection mechanism augmenting the generic DE algorithm (NSODE) to achieve better optimization results. The proposed selection operator adopted in this article aims at a temporary retention of all children generated by the parent generation, and then selecting N better solutions as the new individuals from N parents and N children. Various examples of fuzzy shop scheduling problems are experimented with to test the performance of the improved DE algorithm. The NSODE algorithm is compared with a variety of existing algorithms such as ant colony optimization, particle swarm optimization, and cuckoo search. Experimental results show that the NSODE can obtain superior feasible solutions compared with solutions produced by several algorithms reported in the literature.

Simulation-optimization model for estimating hydraulic conductivity: a numerical case study of the Lu Dila hydropower station in China

A simulation-optimization model (ADINA-MMRDE) has been developed using the commercial finite-element software ADINA coupled with an improved differential evolution algorithm (MMRDE) to estimate the hydraulic conductivity of the aquifer of the Lu Dila hydropower project in China. The ADINA-MMRDE model employed the sum of the weighted square difference between the simulated value and the observed value as the objective function. To evaluate the accuracy of the MMRDE, the ADINA software was also coupled with a classical differential evolution (DE) algorithm and particle swarm optimization (PSO). Before estimating the initial seepage field of the Lu Dila hydropower station, the performance of the ADINA-MMRDE model was first tested using a boundary-simplified model based on the initial seepage field model. The effects of different objective functions, measurement errors and population sizes on the performance of ADINA-MMRDE were investigated. Then the ADINA-MMRDE model was applied to the Lu Dila hydropower project by considering water levels in 20 boreholes. The results showed that the head calculated using the estimated parameters was very close to its observed value. The water levels of 13 monitoring boreholes were used for further confirmation of the parameters obtained by the inversion. The results revealed that the calculated values of the monitoring borehole water levels were generally well fitted to the measured values. This study demonstrated the effectiveness of the ADINA-MMRDE model for practical and complex parameter inversion tasks.

Identification of the Thermal Parameters for the NH gG Fuse Using the Differential Evolution

This paper describes the identification of the thermal parameters for the high breaking capacity low-voltage NH gG fuse. For the process of parameter’s determination, a 3D numerical model of the fuse is used together with the measured temperature on the fuse and differential evolution (DE) as the powerful optimizer for optimization problems. DE’s main task is to reduce the difference between the measured and calculated fuse temperatures. The classic DE algorithm is compared with three improved DE algorithms that have a reduction of the population size during the optimization process. The results of all four algorithms are compared, and the most favorable choice of algorithm is given for such temperature coefficient calculations. The fuse model is appropriate for the calculation of the fusing temperature, according to the excellent agreement between the measured and calculated fuse temperatures.

Trajectory Following and Improved Differential Evolution Solution for Rapid Forming of UAV Formation

In this paper we proposed the circle trajectory assembly algorithm to control the multi-UAVs circular assembly formation. CTFAP solution provides rapid formation of UAVs on a circular orbit and solve the problem of large scattering distance. Proposed Distributed model prediction control framework improves the optimization ability and reduces the computation consumption with the better convergence ability of the UAV formation. Firstly, a circular trajectory following algorithm with an adaptive parameter is proposed to complete the rapid formation of UAVs on a circular orbit and solve the problem of large scattering distance during formation forming. Then, in the stage of formation reconfiguration, with distributed model prediction control framework (DMPC), the proposed method gets the prediction information of DMPC to optimize the population of classical differential evolution (DE) algorithm and improve the iterative optimization ability of DE algorithm. Experiments show that the proposed differential evolution algorithm greatly improves the efficiency of solving the formation reconfiguration problem under the DMPC framework and overcomes the disadvantages of random population of classical DE. For the proposed rapid forming method, assembly range is reduced by 41% compared with direct linearly formation assembly, and the formation forming time is reduced by approximately 21%. Compared with other optimization algorithms such as Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and DE, the proposed differential evolution algorithm reduces instruction response time of single-drones by 16%–30%.