A memory based differential evolution algorithm for unconstrained optimization

In optimization, the performance of differential evolution (DE) and their hybrid versions exist in the literature is highly affected by the inappropriate choice of its operators like mutation and crossover. In general practice, during simulation DE does not employ any strategy of memorizing the so-far-best results obtained in the initial part of the previous generation. In this paper, a new “Memory based DE (MBDE)” presented where two “swarm operators” have been introduced. These operators based on the pBEST and gBEST mechanism of particle swarm optimization. The proposed MBDE is employed to solve 12 basic, 25 CEC 2005, and 30 CEC 2014 unconstrained benchmark functions. In order to further test its efficacy, five different test system of model order reduction (MOR) problem for single-input and single-output system are solved by MBDE. The results of MBDE are compared with state-of-the-art algorithms that also solved those problems. Numerical, statistical, and graphical analysis reveals the competency of the proposed MBDE.     

Water cycle algorithm with evaporation rate for solving constrained and unconstrained optimization problems

This paper presents a modified version of the water cycle algorithm (WCA). The fundamental concepts and ideas which underlie the WCA are inspired based on the observation of water cycle process and how rivers and streams flow to the sea. New concept of evaporation rate for different rivers and streams is defined so called evaporation rate based WCA (ER-WCA), which offers improvement in search. Furthermore, the evaporation condition is also applied for streams that directly flow to sea based on the new approach. The ER-WCA shows a better balance between exploration and exploitation phases compared to the standard WCA. It is shown that the ER-WCA offers high potential in finding all global optima of multimodal and benchmark functions. The WCA and ER-WCA are tested using several multimodal benchmark functions and the obtained optimization results show that in most cases the ER-WCA converges to the global solution faster and offers more accurate results than the WCA and other considered optimizers. Based on the performance of ER-WCA on a number of well-known benchmark functions, the efficiency of the proposed method with respect to the number of function evaluations (computational effort) and accuracy of function value are represented.     

一种改进的约束优化差分进化算法

提出一种改进的差分进化算法用于求解约束优化问题.该算法在处理约束时不引入惩罚因子,使约束处理问题简单化.利用佳点集方法初始化个体以维持种群的多样性.结合差分进化算法两种不同变异策略的特点,对可行个体与不可行个体分别采用DE/best/1变异策略和DE/rand/1策略,以提高算法的全局收敛性能和收敛速率.用几个标准的Benchmark问题进行了测试,实验结果表明该算法是一种求解约束优化问题的有效方法.     

A novel differential evolution algorithm using local abstract convex underestimate strategy for global optimization

Two main challenges in differential evolution (DE) are reducing the number of function evaluations required to obtain optimal solutions and balancing the exploration and exploitation. In this paper, a local abstract convex underestimate strategy based on abstract convexity theory is proposed to address these two problems. First, the supporting hyperplanes are constructed for the neighboring individuals of the trial individual. Consequently, the underestimate value of the trial individual can be obtained by the supporting hyperplanes of its neighboring individuals. Through the guidance of the underestimate value in the select operation, the number of function evaluations can be reduced obviously. Second, some invalid regions of the domain where the global optimum cannot be found are safely excluded according to the underestimate information to improve reliability and exploration efficiency. Finally, the descent directions of supporting hyperplanes are employed for local enhancement to enhance exploitation capability. Accordingly, a novel DE algorithm using local abstract convex underestimate strategy (DELU) is proposed. Numerical experiments on 23 bound-constrained benchmark functions show that the proposed DELU is significantly better than, or at least comparable to several state-of-the art DE variants, non-DE algorithms, and surrogate-assisted evolutionary algorithms.     

基于代理模型的差分进化约束优化

针对耗时计算目标函数的约束优化问题,提出用代理模型来代替耗时计算目标函数的方法,并结合目标函数的信息对约束个体进行选择,从而提出基于代理模型的差分进化约束优化算法。首先,采用拉丁超立方采样方法建立初始种群,用耗时计算目标函数对初始种群进行评估,并以此为样本数据建立目标函数的神经网络代理模型。然后,用差分进化方法为种群中的每一个亲本产生后代,并对后代使用代理模型进行评估,采用可行性规则来比较后代与其亲本并更新种群,根据替换机制将种群中较劣的个体替换为备用存档中较优的个体。最后,当达到最大适应度评估次数时算法停止,给出最优解。该算法与对比算法在10个测试函数上运行的结果表明,该算法得出的结果更精确。将该算法应用于工字梁优化问题的结果表明,相较于优化前的算法,该算法的适应度评估次数减少了80%;相对于FROFI（Feasibility Rule with the incorporation of Objective Function Information）算法,该算法的适应度评估次数减少了36%。运用所提算法进行优化可以有效减少调用耗时计算目标函数的次数,提升优化效率,节约计算成本。     

改进差异演化算法求解约束优化问题

在现实生活中许多实际问题都可以转化为约束优化问题,并且实际问题通常都很复杂,其函数形态各具特色,传统基于梯度信息的各种求解策略对于具有不可微、多峰及非凸的非线性函数约束优化问题很难凑效。而最近兴起的智能类算法却对这类问题的求解效果突出,在借鉴国外的差异演化算法研究成果基础上,运用改进差异演化算法来求解约束优化问题。最后通过实例进行仿真实验,结果表明改进差异演化算法在求解约束优化问题时具有一定的优越性。     

一种改进的多目标约束优化差分进化算法

提出一种新的多目标优化差分进化算法用于求解约束优化问题.该算法利用佳点集方法初始化个体以维持种群的多样性.将约束优化问题转化为两个目标的多目标优化问题.基于Pareto支配关系,将种群分为Pareto子集和Non-Pareto子集,结合差分进化算法两种不同变异策略的特点,对Non-Pareto子集和Pareto子集分别采用DE/best/1变异策略和DE/rand/1变异策略.数值实验结果表明该算法具有较好的寻优效果.     

An improved constrained differential evolution algorithm for unmanned aerial vehicle global route planning

This paper formulates the global route planning problem for the unmanned aerial vehicles (UAVs) as a constrained optimization problem in the three-dimensional environment and proposes an improved constrained differential evolution (DE) algorithm to generate an optimal feasible route. The flight route is designed to have a short length and a low flight altitude. The multiple constraints based on the realistic scenarios are taken into account, including maximum turning angle, maximum climbing/gliding slope, terrain, forbidden flying areas, map and threat area constraints. The proposed DE-based route planning algorithm combines the standard DE with the level comparison method and an improved strategy is proposed to control the satisfactory level. To show the high performance of the proposed method, we compare the proposed algorithm with six existing constrained optimization algorithms and five penalty function based methods. Numerical experiments in two test cases are carried out. Our proposed algorithm demonstrates a good performance in terms of the solution quality, robustness, and the constraint-handling ability.     

An improved differential evolution algorithm with triangular mutation for global numerical optimization

This paper presents an Improved Differential Evolution (IDE) algorithm for solving global numerical optimization problems over continuous space. The proposed algorithm introduces a new triangular mutation rule based on the convex combination vector of the triplet defined by the three randomly chosen vectors and the difference vector between the best and the worst individuals among the three randomly selected vectors. The mutation rule is combined with the basic mutation strategy through a non-linear decreasing probability rule. Furthermore, a restart mechanism is also proposed to avoid premature convergence. IDE is tested on a well-known set of unconstrained problems and shows its superiority to state-of-the-art differential evolution variants.     

非线性约束优化问题的自适应差分进化算法

提出了一种非线性约束优化问题改进的自适应差分进化算法。该算法对差分进化算法中固定的加权因子和交叉概率因子进行改进;定义了约束违反度函数,将约束优化问题转化为无约束双目标优化问题,在每次迭代中按照约束违反度的大小保留一部分性能较优不可行粒子,有效地维持了种群的多样性;为了扩大粒子的搜索范围引入变异算子。数值实验表明,新算法具有较快的收敛速度和较好的全局寻优能力。     

Neighborhood-adaptive differential evolution for global numerical optimization

In this study, we consider the scenario that differential evolution (DE) is applied for global numerical optimization and the index-based neighborhood information of population is used for enhancing the performance of DE. Although many methods are developed under this scenario, neighborhood information of current population has not been systematically exploited in the DE algorithm design. Furthermore, previous studies have shown the effect of neighborhood topology interacted with the function being solved. However, there are few investigations of DE that consider different topologies for different functions during the evolutionary process. Motivated by these observations, a new DE framework, named neighborhood-adaptive DE (NaDE), is presented. In NaDE, a pool of index-based neighborhood topologies is firstly used to define multiple neighborhood relationships for each individual and then the neighborhood relationships are adaptively selected for the specific functions during the evolutionary process. In this way, a more appropriate neighborhood relationship for each individual can be determined adaptively to match different phases of the search process for the function being solved. After that, a neighborhood-dependent directional mutation operator is introduced into NaDE to generate a new solution with the selected neighborhood topology. Being a general framework, NaDE is easy to implement and can be realized with most existing DE algorithms. In order to test the effectiveness of the proposed framework, we have evaluated NaDE via investigating several instantiations of it. Experimental results have shown that NaDE generally outperforms its corresponding DE algorithm on different kinds of optimization problems. Moreover, the synergy among different neighborhood topologies in NaDE is also revealed when compared with the DE variants with single neighborhood topology.     

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.     

Differential evolution with guiding archive for global numerical optimization

Differential evolution (DE) is a simple, yet efficient, population-based global evolutionary algorithm. DE may suffer from stagnation. This study presents a DE framework with guiding archive (GAR-DE) to help DE escape from the situation of stagnation. The proposed framework constructs a guiding archive and executes stagnation detection at each iteration. Guiding archive is composed of a certain number of relatively high-quality solutions. These solutions are collected in terms of fitness as well as diversity. If a stagnated individual is detected, the proposed framework selects a solution from guiding archive to replace the base vector in mutation operator. In this way, more promising solutions are provided to guide the evolution and effectively help DE escape from the situation of stagnation. The proposed framework is applied to six original DE algorithms, as well as two advanced DE variants. Experimental results on 28 benchmark functions and 8 real-world application problems show that the proposed framework can enhance the performance of most DE algorithms studied.     

具有混沌局部搜索策略的差分进化全局优化算法

提出了一种具有混沌局部搜索策略的差分进化全局优化算法（CLSDE）,它是在每一代中通过DE/best/1/bin形式的差分进化算法找到最佳个体,然后在最佳个体的附近用混沌的方法进行局部搜索。8个基本的测试函数优化结果表明：若误差函数精度为10^-10,CLSDE寻优成功率比DE和SACDE都要高,而且收敛速度比DE和SACDE都要快。     

一种求解非线性约束优化全局最优的新方法

本文提出了一种求解非线性约束优化的全局最优的新方法—它是基于利用非线性互补函数和不断增加新的约束来重复解库恩-塔克条件的非线性方程组的新方法。因为库恩-塔克条件是非线性约束优化的必要条件,得到的解未必是非线性约束优化的全局最优解,为此,本文首次给出了通过利用该优化问题的先验知识,不断地增加约束来限制全局最优解范围的方法,一些仿真例子表明提出的方法和理论有效的,并且可行的。     

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.     

求解约束优化的一个自适应杂交差分演化算法

结合基于可行性规则的约束处理技术,构造了一个求解约束优化问题的自适应杂交差分演化模拟退火算法。该算法以差分演化算法为基础,用模拟退火策略来增强种群的多样性,用一个基于可行性规则的约束处理技术来处理不等式约束,且自适应化关键控制参数,避开人为控制参数的困难。在标准测试集上的实验结果表明该算法的有效性,与同类算法的比较表明了该算法的优越性。     

A hybrid differential evolution augmented Lagrangian method for constrained numerical and engineering optimization

We present a new hybrid method for solving constrained numerical and engineering optimization problems in this paper. The proposed hybrid method takes advantage of the differential evolution (DE) ability to find global optimum in problems with complex design spaces while directly enforcing feasibility of constraints using a modified augmented Lagrangian multiplier method. The basic steps of the proposed method are comprised of an outer iteration, in which the Lagrangian multipliers and various penalty parameters are updated using a first-order update scheme, and an inner iteration, in which a nonlinear optimization of the modified augmented Lagrangian function with simple bound constraints is implemented by a modified differential evolution algorithm. Experimental results based on several well-known constrained numerical and engineering optimization problems demonstrate that the proposed method shows better performance in comparison to the state-of-the-art algorithms.     

基于微分进化的混合生物地理学约束优化算法

针对生物地理学优化算法（biogeography based optimization ,BBO）容易陷入局部最优解的缺点,提出一种基于微分进化（differential evolution ,DE）改进BBO算法的混合生物地理学（BBO‐DE）优化算法。通过有机结合BBO算法的利用能力和DE算法的搜索能力,实现利用能力与搜索能力的平衡;引入基于可行性的约束处理机制,解决传统BBO算法无法求解约束优化的问题。通过选定的8个标准测试函数对改进算法进行仿真测试,测试结果验证了改进算法的可行性和有效性,与基本BBO和DE算法相比,其在最终解的质量和收敛速度上具有明显优势。     

A hybrid shuffled complex evolution approach with pattern search for unconstrained optimization

The difficulties associated with using classical mathematical programming methods on complex optimization problems have contributed to the development of alternative and efficient numerical approaches. Recently, to overcome the limitations of classical optimization methods, researchers have proposed a wide variety of meta-heuristics for searching near-optimum solutions to problems. Among the existing meta-heuristic algorithms, a relatively new optimization paradigm is the Shuffled Complex Evolution at the University of Arizona (SCE-UA) which is a global optimization strategy that combines concepts of the competition evolution theory, downhill simplex procedure of Nelder-Mead, controlled random search and complex shuffling. In an attempt to reduce processing time and improve the quality of solutions, particularly to avoid being trapped in local optima, in this paper is proposed a hybrid SCE-UA approach. The proposed hybrid algorithm is the combination of SCE-UA (without Nelder-Mead downhill simplex procedure) and a pattern search approach, called SCE-PS, for unconstrained optimization. Pattern search methods are derivative-free, meaning that they do not use explicit or approximate derivatives. Moreover, pattern search algorithms are direct search methods well suitable for the global optimization of highly nonlinear, multiparameter, and multimodal objective functions. The proposed SCE-PS method is tested with six benchmark optimization problems. Simulation results show that the proposed SCE-PS improves the searching performance when compared with the classical SCE-UA and a genetic algorithm with floating-point representation for all the tested problems. As evidenced by the performance indices based on the mean performance of objective function in 30 runs and mean of computational time, the SCE-PS algorithm has demonstrated to be effective and efficient at locating best-practice optimal solutions for unconstrained optimization.