Constrained differential evolution using generalized opposition-based learning

Differential evolution (DE) is a well-known optimization approach to deal with nonlinear and complex optimization problems. However, many real-world optimization problems are constrained problems that involve equality and inequality constraints. DE with constraint handling techniques, named constrained differential evolution (CDE), can be used to solve constrained optimization problems. In this paper, we propose a new CDE framework that uses generalized opposition-based learning (GOBL), named GOBL-CDE. In GOBL-CDE, firstly, the transformed population is generated using general opposition-based learning in the population initialization. Secondly, the transformed population and the initial population are merged and only half of the best individuals are selected to compose the new initial population to proceed mutation, crossover, and selection. Lastly, based on a jumping probability, the transformed population is calculated again after generating new populations, and the fittest individuals are selected to compose new population from the union of the current population and the transformed population. The GOBL-CDE framework can be applied to most CDE variants. As examples, in this study, the framework is applied to two popular representative CDE variants, i.e., rank-iMDDE and \(\varepsilon \)DEag. Experiment results on 24 benchmark functions from CEC’2006 and 18 benchmark functions from CEC’2010 show that the proposed framework is an effective approach to enhance the performance of CDE algorithms.     

Surrogate assisted‐hybrid differential evolution algorithm using diversity control

Evolutionary algorithms (EAs) being a major optimization framework, typically require a considerable number of function evaluations to locate an optimal solution for computationally intensive real‐world optimization problems. In order to solve complex multimodal problems within a limited computational budget, surrogate models are integrated with EA. The overall performance of such algorithms not only depends on the construction and integration procedure of the model, but also on the efficiency of the underlying EA in overcoming premature convergence. This can be achieved through diversity control and parameter adaptation methodology in EAs. In this paper, an improved algorithm, namely Diversity Controlled Parameter adapted Differential Evolution with Local Search (DCPaDE‐LS) is integrated into two dynamic surrogate models and two variants, namely Surrogate Assisted Parameter adapted Differential Evolution with Artificial Neural Networks and Response Surface Methodology (SAPDE‐ANN and SAPDE‐RSM) are proposed. They reduce the exact function evaluations for complex, multimodal problems. The performance of the proposed variants are compared on a set of 26 bound‐constrained benchmark functions scalable with 10 and 30 dimensions, with respect to average number of function evaluations (NFE), success rate (SR) and % reduction in NFE in 30 independent trials. The SAPDE variants are compared with Self‐adaptive Differential Evolution, DCPaDE‐LS, Increasing Population Size Covariance Matrix Adaptation Evolution Strategy and Comprehensive Learning Particle Swarm Optimization. The SAPDE variants are able to reduce the NFE without loss in SR for all the functions. The algorithms are also validated using 12 solvable functions from CEC 2005. Of the two variants, SAPDE‐ANN reports reduced NFE in more functions compared with SAPDE‐RSM. Results reveal that the proposed SAPDE algorithm can be applied to real‐world optimization problems.     

一种改进的基于差分进化的多目标进化算法

近年来运用进化算法（EAs）解决多目标优化问题（Multi-objective Optimization Problems MOPs）引起了各国学者们的关注。作为一种基于种群的优化方法,EAs提供了一种在一次运行后得到一组优化的解的方法。差分进化（DE）算法是EA的一个分支,最开始是用来解决连续函数空间的问题。提出了一种改进的基于差分进化的多目标进化算法（CDE）,并且将它与另外两个经典的多目标进化算法（MOEAs）NSGA-II和SPEA2进行了对比实验。     

A space search optimization algorithm with accelerated convergence strategies

Evolutionary algorithms (EAs), which have been widely used to solve various scientific and engineering optimization problems, are essentially stochastic search algorithms operating in the overall solution space. However, such random search mechanism may lead to some disadvantages such as a long computing time and premature convergence. In this study, we propose a space search optimization algorithm (SSOA) with accelerated convergence strategies to alleviate the drawbacks of the purely random search mechanism. The overall framework of the SSOA involves three main search mechanisms: local space search, global space search, and opposition-based search. The local space search that aims to form new solutions approaching the local optimum is realized based on the concept of augmented simplex method, which exhibits significant search abilities realized in some local space. The global space search is completed by Cauchy searching, where the approach itself is based on the Cauchy mutation. This operation can help the method avoid of being trapped in local optima and in this way alleviate premature convergence. An opposition-based search is exploited to accelerate the convergence of space search. This operator can effectively reduce a substantial computational overhead encountered in evolutionary algorithms (EAs). With the use of them SSOA realizes an effective search process. To evaluate the performance of the method, the proposed SSOA is contrasted with a method of differential evolution (DE), which is a well-known space concept-based evolutionary algorithm. When tested against benchmark functions, the SSOA exhibits a competitive performance vis-a-vis performance of some other competitive schemes of differential evolution in terms of accuracy and speed of convergence, especially in case of high-dimensional continuous optimization problems.     

Hybrid adaptive evolutionary algorithm based on decomposition

The performance of search operators varies across the different stages of the search/optimization process of evolutionary algorithms (EAs). In general, a single search operator may not do well in all these stages when dealing with different optimization and search problems. To mitigate this, adaptive search operator schemes have been introduced. The idea is that when a search operator hits a difficult patch (under-performs) in the search space, the EA scheme “reacts” to that by potentially calling upon a different search operator. Hence, several multiple-search operator schemes have been proposed and employed within EA. In this paper, a hybrid adaptive evolutionary algorithm based on decomposition (HAEA/D) that employs four different crossover operators is suggested. Its performance has been evaluated on the well-known IEEE CEC’09 test instances. HAEA/D has generated promising results which compare well against several well-known algorithms including MOEA/D, on a number of metrics such as the inverted generational distance (IGD), the hyper-volume, the Gamma and Delta functions. These results are included and discussed in this paper.     

Opposition-Based Differential Evolution

Evolutionary algorithms (EAs) are well-known optimization approaches to deal with nonlinear and complex problems. However, these population-based algorithms are computationally expensive due to the slow nature of the evolutionary process. This paper presents a novel algorithm to accelerate the differential evolution (DE). The proposed opposition-based DE (ODE) employs opposition-based learning (OBL) for population initialization and also for generation jumping. In this work, opposite numbers have been utilized to improve the convergence rate of DE. A comprehensive set of 58 complex benchmark functions including a wide range of dimensions is employed for experimental verification. The influence of dimensionality, population size, jumping rate, and various mutation strategies are also investigated. Additionally, the contribution of opposite numbers is empirically verified. We also provide a comparison of ODE to fuzzy adaptive DE (FADE). Experimental results confirm that the ODE outperforms the original DE and FADE in terms of convergence speed and solution accuracy.     

A memory efficient stochastic evolution based algorithm for the multi-objective shortest path problem

Multi-objective shortest path (MOSP) problem aims to find the shortest path between a pair of source and a destination nodes in a network. This paper presents a stochastic evolution (StocE) algorithm for solving the MOSP problem. The proposed algorithm is a single-solution-based evolutionary algorithm (EA) with an archive for storing several non-dominant solutions. The solution quality of the proposed algorithm is comparable to the established population-based EAs. In StocE, the solution replaces its bad characteristics as the generations evolve. In the proposed algorithm, different sub-paths are the characteristics of the solution. Using the proposed perturb operation, it eliminates the bad sub-paths from generation to generation. The experiments were conducted on huge real road networks. The proposed algorithm is comparable to well-known single-solution and population-based EAs. The single-solution-based EAs are memory efficient, whereas, the population-based EAs are known for their good solution quality. The performance measures were the solution quality, speed and memory consumption, assessed by the hypervolume (HV) metric, total number of evaluations and memory requirements in megabytes. The HV metric of the proposed algorithm is superior to that of the existing single-solution and population-based EAs. The memory requirements of the proposed algorithm is at least half than the EAs delivering similar solution quality. The proposed algorithms also executes more rapidly than the existing single-solution-based algorithms. The experimental results show that the proposed algorithm is suitable for solving MOSP problems in embedded systems.     

Constrained differential evolution with multiobjective sorting mutation operators for constrained optimization

Differential evolution (DE) is a simple and powerful evolutionary algorithm for global optimization. DE with constraint handling techniques, named constrained differential evolution (CDE), can be used to solve constrained optimization problems (COPs). In existing CDEs, the parents are randomly selected from the current population to produce trial vectors. However, individuals with fitness and diversity information should have more chances to be selected. This study proposes a new CDE framework that uses nondominated sorting mutation operator based on fitness and diversity information, named MS-CDE. In MS-CDE, firstly, the fitness of each individual in the population is calculated according to the current population situation. Secondly, individuals in the current population are ranked according to their fitness and diversity contribution. Lastly, parents in the mutation operators are selected in proportion to their rankings based on fitness and diversity. Thus, promising individuals with better fitness and diversity are more likely to be selected as parents. The MS-CDE framework can be applied to most CDE variants. In this study, the framework is applied to two popular representative CDE variants, (μ + λ)-CDE and ECHT-DE. Experiment results on 24 benchmark functions from CEC’2006 and 18 benchmark functions from CEC’2010 show that the proposed framework is an effective approach to enhance the performance of CDE algorithms.     

Drift conditions for estimating the first hitting times of evolutionary algorithms

For the global optimization problems with continuous variables, evolutionary algorithms (EAs) are often used to find the approximate solutions. The number of generations for an EA to find the approximate solutions, called the first hitting time, is an important index to measure the performance of the EA. However, calculating the first hitting time is still difficult in theory. This paper proposes some new drift conditions that are used to estimate the upper bound of the first hitting times of EAs for finding the approximate solutions. Two case studies are given to show how to apply these conditions to estimate the first hitting times.     

Integrating Variable Reduction Strategy With Evolutionary Algorithms for Solving Nonlinear Equations Systems

Nonlinear equations systems (NESs) are widely used in real-world problems and they are difficult to solve due to their nonlinearity and multiple roots. Evolutionary algorithms (EAs) are one of the methods for solving NESs, given their global search capabilities and ability to locate multiple roots of a NES simultaneously within one run. Currently, the majority of research on using EAs to solve NESs focuses on transformation techniques and improving the performance of the used EAs. By contrast, problem domain knowledge of NESs is investigated in this study, where we propose the incorporation of a variable reduction strategy (VRS) into EAs to solve NESs. The VRS makes full use of the systems of expressing a NES and uses some variables (i.e., core variable) to represent other variables (i.e., reduced variables) through variable relationships that exist in the equation systems. It enables the reduction of partial variables and equations and shrinks the decision space, thereby reducing the complexity of the problem and improving the search efficiency of the EAs. To test the effectiveness of VRS in dealing with NESs, this paper mainly integrates the VRS into two existing state-of-the-art EA methods (i.e., MONES and DR-JADE) according to the integration framework of the VRS and EA, respectively. Experimental results show that, with the assistance of the VRS, the EA methods can produce better results than the original methods and other compared methods. Furthermore, extensive experiments regarding the influence of different reduction schemes and EAs substantiate that a better EA for solving a NES with more reduced variables tends to provide better performance.      

Parallelism and evolutionary algorithms

This paper contains a modern vision of the parallelization techniques used for evolutionary algorithms (EAs). The work is motivated by two fundamental facts: 1) the different families of EAs have naturally converged in the last decade while parallel EAs (PEAs) are still lack of unified studies; and 2) there is a large number of improvements in these algorithms and in their parallelization that raise the need for a comprehensive survey. We stress the differences between the EA model and its parallel implementation throughout the paper. We discuss the advantages and drawbacks of PEAs. Also, successful applications are mentioned and open problems are identified. We propose potential solutions to these problems and classify the different ways in which recent results in theory and practice are helping to solve them. Finally, we provide a highly structured background relating to PEAs in order to make researchers aware of the benefits of decentralizing and parallelizing an EA     

Parameter control system of evolutionary algorithm that is aided by the entire search history

In solving problems with evolutionary algorithms (EAs), the performance of the EA will be affected by its properties. As the properties of EA depend on the parameter setting, users need to tune the parameters to optimize the performance on different problems. In the case that the user does not have any prior knowledge of the problem, parameter tuning is very difficult and time consuming. One needs to try different combinations of parameter values to find the best setting. To solve this problem, one way is to control the parameters during the EA run. This paper proposes a new adaptive parameter control system, called Parameter Control system using entire Search History (PCSH). It is a general add-on system which is not restricted to a specific class of EA. Users are only required to know the range of the parameters. It automatically adjusts the parameters of an EA according to the entire search history, in a parameter-less manner. To illustrate the performance of PCSH, it is applied to control the parameters of three common classes of EAs: (1) canonical Genetic Algorithm (GA), (2) Particle Swarm Optimization (PSO) and (3) Differential Evolution (DE). For GA, we show that PCSH can automatically control the crossover operator, crossover values (uniformly sampled from the range) and mutation operator. For DE, we show that PCSH can automatically control the crossover operator, crossover values and the differential amplification factor (uniformly sampled from the ranges). For PSO, we show that PCSH can automatically control the two learning factors and the inertia weight (uniformly sampled from the range). Moreover, no special provision is needed at the initialization. 34 benchmark functions are used to evaluate the performance comprehensively. The test results show that, in most of the benchmark functions, the performance of the test EAs are improved or similar after adopting PCSH. It shows that PCSH keeps or improves the performance of the test EAs while relieving the heavy burden of the algorithm designer on the setting of some parameters.     

An impatient evolutionary algorithm with probabilistic tabu search for unified solution of some NP-hard problems in graph and set theory via clique finding.

Many graph- and set-theoretic problems, because of their tremendous application potential and theoretical appeal, have been well investigated by the researchers in complexity theory and were found to be NP-hard. Since the combinatorial complexity of these problems does not permit exhaustive searches for optimal solutions, only near-optimal solutions can be explored using either various problem-specific heuristic strategies or metaheuristic global-optimization methods, such as simulated annealing, genetic algorithms, etc. In this paper, we propose a unified evolutionary algorithm (EA) to the problems of maximum clique finding, maximum independent set, minimum vertex cover, subgraph and double subgraph isomorphism, set packing, set partitioning, and set cover. In the proposed approach, we first map these problems onto the maximum clique-finding problem (MCP), which is later solved using an evolutionary strategy. The proposed impatient EA with probabilistic tabu search (IEA-PTS) for the MCP integrates the best features of earlier successful approaches with a number of new heuristics that we developed to yield a performance that advances the state of the art in EAs for the exploration of the maximum cliques in a graph. Results of experimentation with the 37 DIMACS benchmark graphs and comparative analyses with six state-of-the-art algorithms, including two from the smaller EA community and four from the larger metaheuristics community, indicate that the IEA-PTS outperforms the EAs with respect to a Pareto-lexicographic ranking criterion and offers competitive performance on some graph instances when individually compared to the other heuristic algorithms. It has also successfully set a new benchmark on one graph instance. On another benchmark suite called Benchmarks with Hidden Optimal Solutions, IEA-PTS ranks second, after a very recent algorithm called COVER, among its peers that have experimented with this suite.     

Evolutionary design of Evolutionary Algorithms

Manual design of Evolutionary Algorithms (EAs) capable of performing very well on a wide range of problems is a difficult task. This is why we have to find other manners to construct algorithms that perform very well on some problems. One possibility (which is explored in this paper) is to let the evolution discover the optimal structure and parameters of the EA used for solving a specific problem. To this end a new model for automatic generation of EAs by evolutionary means is proposed here. The model is based on a simple Genetic Algorithm (GA). Every GA chromosome encodes an EA, which is used for solving a particular problem. Several Evolutionary Algorithms for function optimization are generated by using the considered model. Numerical experiments show that the EAs perform similarly and sometimes even better than standard approaches for several well-known benchmarking problems.     

Population Markov Chain Monte Carlo

Stochastic search algorithms inspired by physical and biological systems are applied to the problem of learning directed graphical probability models in the presence of missing observations and hidden variables. For this class of problems, deterministic search algorithms tend to halt at local optima, requiring random restarts to obtain solutions of acceptable quality. We compare three stochastic search algorithms: a Metropolis-Hastings Sampler (MHS), an Evolutionary Algorithm (EA), and a new hybrid algorithm called Population Markov Chain Monte Carlo, or popMCMC. PopMCMC uses statistical information from a population of MHSs to inform the proposal distributions for individual samplers in the population. Experimental results show that popMCMC and EAs learn more efficiently than the MHS with no information exchange. Populations of MCMC samplers exhibit more diversity than populations evolving according to EAs not satisfying physics-inspired local reversibility conditions.     

A Multiagent Evolutionary Algorithm for Combinatorial Optimization Problems

Based on our previous works, multiagent systems and evolutionary algorithms (EAs) are integrated to form a new algorithm for combinatorial optimization problems (CmOPs), namely, MultiAgent EA for CmOPs (MAEA-CmOPs). In MAEA-CmOPs, all agents live in a latticelike environment, with each agent fixed on a lattice point. To increase energies, all agents compete with their neighbors, and they can also increase their own energies by making use of domain knowledge. Theoretical analyses show that MAEA-CmOPs converge to global optimum solutions. Since deceptive problems are the most difficult CmOPs for EAs, in the experiments, various deceptive problems with strong linkage, weak linkage, and overlapping linkage, and more difficult ones, namely, hierarchical problems with treelike structures, are used to validate the performance of MAEA-CmOPs. The results show that MAEA-CmOP outperforms the other algorithms and has a fast convergence rate. MAEA-CmOP is also used to solve large-scale deceptive and hierarchical problems with thousands of dimensions, and the experimental results show that MAEA-CmOP obtains a good performance and has a low computational cost, which the time complexity increases in a polynomial basis with the problem size.      

Unveiling evolutionary algorithm representation with DU maps

Evolutionary algorithms (EAs) have proven to be effective in tackling problems in many different domains. However, users are often required to spend a significant amount of effort in fine-tuning the EA parameters in order to make the algorithm work. In principle, visualization tools may be of great help in this laborious task, but current visualization tools are either EA-specific, and hence hardly available to all users, or too general to convey detailed information. In this work, we study the Diversity and Usage map (DU map), a compact visualization for analyzing a key component of every EA, the representation of solutions. In a single heat map, the DU map visualizes for entire runs how diverse the genotype is across the population and to which degree each gene in the genotype contributes to the solution. We demonstrate the generality of the DU map concept by applying it to six EAs that use different representations (bit and integer strings, trees, ensembles of trees, and neural networks). We present the results of an online user study about the usability of the DU map which confirm the suitability of the proposed tool and provide important insights on our design choices. By providing a visualization tool that can be easily tailored by specifying the diversity (D) and usage (U) functions, the DU map aims at being a powerful analysis tool for EAs practitioners, making EAs more transparent and hence lowering the barrier for their use.     

融合改进天牛须搜索的教与学优化算法

针对教与学优化算法易早熟,解精度低,甚至收敛于局部最优的问题,提出一种新的融合改进天牛须搜索的教与学优化算法.该算法利用Tent映射反向学习策略初始化种群,提升初始解质量.在"教"阶段,对教师个体执行天牛须搜索算法,增强教师教学水平,提高最优解的精确性.在"学"阶段,对学生个体进行混合变异,从而跳出局部最优,平衡算法的...     

On the equivalences and differences of evolutionary algorithms

Evolutionary algorithms (EAs) are fast and robust computation methods for global optimization, and have been widely used in many real-world applications. We first conceptually discuss the equivalences of various popular EAs including genetic algorithm (GA), biogeography-based optimization (BBO), differential evolution (DE), evolution strategy (ES) and particle swarm optimization (PSO). We find that the basic versions of BBO, DE, ES and PSO are equal to the GA with global uniform recombination (GA/GUR) under certain conditions. Then we discuss their differences based on biological motivations and implementation details, and point out that their distinctions enhance the diversity of EA research and applications. To further study the characteristics of various EAs, we compare the basic versions and advanced versions of GA, BBO, DE, ES and PSO to explore their optimization ability on a set of real-world continuous optimization problems. Empirical results show that among the basic versions of the algorithms, BBO performs best on the benchmarks that we studied. Among the advanced versions of the algorithms, DE and ES perform best on the benchmarks that we studied. However, our main conclusion is that the conceptual equivalence of the algorithms is supported by the fact that algorithmic modifications result in very different performance levels.     

基于反向学习的自适应差分进化算法

为解决差分进化（DE）算法过早收敛与搜索能力低的问题,讨论对控制参数的动态调整,提出一种基于反向学习的自适应差分进化算法。该算法通过反向精英学习机制来增强种群的局部搜索能力,获取精确度更高的最优个体;同时,采用高斯分布随机性提高单个个体的开发能力,通过扩充种群的多样性,避免算法过早收敛,整体上平衡全局搜索与局部寻优的能力。采用CEC 2014中的6个测试函数进行仿真实验,并与其他差分进化算法进行对比,实验结果表明所提算法在收敛速度、收敛精度及可靠性上表现更优。