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.     

Probabilistic properties of fitness-based quasi-reflection in evolutionary algorithms

Evolutionary algorithms (EAs) excel in optimizing systems with a large number of variables. Previous mathematical and empirical studies have shown that opposition-based algorithms can improve EA performance. We review existing opposition-based algorithms and introduce a new one. The proposed algorithm is named fitness-based quasi-reflection and employs the relative fitness of solution candidates to generate new individuals. We provide the probabilistic analysis to prove that among all the opposition-based methods that we investigate, fitness-based quasi-reflection has the highest probability of being closer to the solution of an optimization problem. We support our theoretical findings via Monte Carlo simulations and discuss the use of different reflection weights. We also demonstrate the benefits of fitness-based quasi-reflection on three state-of-the-art EAs that have competed at IEEE CEC competitions. The experimental results illustrate that fitness-based quasi-reflection enhances EA performance, particularly on problems with more challenging solution spaces. We found that competitive DE (CDE) which was ranked tenth in CEC 2013 competition benefited the most from opposition. CDE with fitness-based quasi-reflection improved on 21 out of the 28 problems in the CEC 2013 test suite and achieved 100% success rate on seven more problems than CDE.     

保存基因的2-Opt一般反向差分演化算法

为了进一步提高差分演化算法的性能,提出一种采用保存基因的2-Opt一般反向差分演化算法,并把它应用于函数优化问题中.新算法具有以下特征:(1)采用保存被选择个体基因的方式组成参加演化的新个体.保存基因的方法可以很好的保持种群多样性;(2)采用一般反向学习(GOBL)机制进行初始化,提高了初始化效率;(3)采用2-Opt算法加速差分演化算法的收敛速度,提高搜索效率.通过测试函数的实验,并与其他差分演化算法进行比较.实验结果证实了新算法的高效性,通用性和稳健性.     

Opposition-based learning in the shuffled differential evolution algorithm

This paper proposes using the opposition-based learning (OBL) strategy in the shuffled differential evolution (SDE). In the SDE, population is divided into several memeplexes and each memeplex is improved by the differential evolution (DE) algorithm. The OBL by comparing the fitness of an individual to its opposite and retaining the fitter one in the population accelerates search process. The objective of this paper is to introduce new versions of the DE which, on one hand, use the partitioning and shuffling concepts of SDE to compensate for the limited amount of search moves of the original DE and, on the other hand, employ the OBL to accelerate the DE without making premature convergence. Four versions of DE algorithm are proposed based on the OBL and SDE strategies. All algorithms similarly use the opposition-based population initialization to achieve fitter initial individuals and their difference is in applying opposition-based generation jumping. Experiments on 25 benchmark functions designed for the special session on real-parameter optimization of CEC2005 and non-parametric analysis of obtained results demonstrate that the performances of the proposed algorithms are better than the SDE. The fourth version of proposed algorithm has a significant difference compared to the SDE in terms of all considered aspects. The emphasis of comparison results is to obtain some successful performances on unsolved functions for the first time, which so far have not been reported any successful runs on them. In a later part of the comparative experiments, performance comparisons of the proposed algorithm with some modern DE algorithms reported in the literature confirm a significantly better performance of our proposed algorithm, especially on high-dimensional functions.     

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...     

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.     

求解大规模优化问题的正交反向混合差分进化算法*

差分进化算法简单高效,然而在求解大规模优化问题时,其求解性能迅速降低。针对该问题,提出一种正交反向差分进化算法。首先,该算法利用正交交叉算子,加强了算法的局部搜索能力。其次,为防止过强的局部搜索使算法陷入早熟收敛,利用反向学习策略调节种群多样性,从而有效地平衡算法的全局和局部搜索能力。利用11个标准测试函数进行实验,并和差分进化算法的4种优秀改进版本进行比较,实验结果表明该算法求解精度高、收敛速率快,是一种求解大规模优化问题的有效算法。     

An adaptive surrogate assisted differential evolutionary algorithm for high dimensional constrained problems

Differential evolution (DE) is a competitive algorithm for constrained optimization problems (COPs). In this study, in order to improve the efficiency and accuracy of the DE for high dimensional problems, an adaptive surrogate assisted DE algorithm, called ASA-DE is suggested. In the ASA, several kinds of surrogate modeling techniques are integrated. Furthermore, to avoid violate the constraints and obtain better solution simultaneously, adaptive strategies for population size and mutation are also suggested in this study. The suggested adaptive population strategy which controls the exploring and exploiting states according to whether algorithm find enough feasible solution is similar to a state switch. The mutation strategy is used to enhance the effect of state switch based on adaptive population size. Finally, the suggested ASA-DE is evaluated on the benchmark problems from congress on evolutionary computation (CEC) 2017 constrained real parameter optimization. The experimental results show the proposed algorithm is a competitive one compared to other state-of-the-art algorithms.     

应用反向学习和差分进化的群搜索优化算法

针对标准群搜索优化算法在解决一些复杂优化问题时容易陷入局部最优且收敛速度较慢的问题,提出一种应用反向学习和差分进化的群搜索优化算法(Group Search Optimization with Opposition-based Learning and Diffe-rential Evolution,OBDGSO)。该算法利用一般动态反向学习机制产生反向种群,扩大算法的全局勘探范围;对种群中较优解个体实施差分进化的变异操作,实现在较优解附近的局部开采,以改善算法的求解精度和收敛速度。这两种策略在GSO算法中相互协同,以更好地平衡算法的全局搜索能力和局部开采能力。将OBDGSO算法和另外4种群智能算法在12个基准测试函数上进行实验,结果表明OBDGSO算法在求解精度和收敛速度上具有较显著的性能优势。     

Probabilistic opposition-based particle swarm optimization with velocity clamping

A probabilistic opposition-based Particle Swarm Optimization algorithm with Velocity Clamping and inertia weights (OvcPSO) is designed for function optimization—to accelerate the convergence speed and to optimize solution’s accuracy on standard benchmark functions. In this work, probabilistic opposition-based learning for particles is incorporated with PSO to enhance the convergence rate—it uses velocity clamping and inertia weights to control the position, speed and direction of particles to avoid premature convergence. A comprehensive set of 58 complex benchmark functions including a wide range of dimensions have been used for experimental verification. It is evident from the results that OvcPSO can deal with complex optimization problems effectively and efficiently. A series of experiments have been performed to investigate the influence of population size and dimensions upon the performance of different PSO variants. It also outperforms FDR-PSO, CLPSO, FIPS, CPSO-H and GOPSO on various benchmark functions. Last but not the least, OvcPSO has also been compared with opposition-based differential evolution (ODE); it outperforms ODE on lower swarm population and higher-dimensional functions.     

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。     

基于随机邻域变异和趋优反向学习的差分进化算法

传统差分进化（DE）算法在迭代过程中不能充分平衡全局勘探与局部开发,存在易陷入局部最优、求解精度低、收敛速度慢等缺点。为提升算法性能,提出一种基于随机邻域变异和趋优反向学习的差分进化（RNODE）算法并对其进行复杂度分析。首先,为种群中每个个体生成随机邻域,用全局最佳个体引导邻域最佳个体生成复合基向量,结合控制参数自适应更新机制构成随机邻域变异策略,使算法在引导种群向最优方向趋近的同时保持一定的勘探能力;其次,为了进一步帮助算法跳出局部最优,对种群中较差个体执行趋优反向学习操作,扩大搜索区域;最后,将RNODE与九种算法进行对比以验证RNODE的有效性和先进性。在23个Benchmark函数和两个实际工程优化问题上的实验结果表明,RNODE算法收敛精度更高、速度更快、稳定性更优。     

Differential evolution in constrained numerical optimization: An empirical study

Motivated by the recent success of diverse approaches based on differential evolution (DE) to solve constrained numerical optimization problems, in this paper, the performance of this novel evolutionary algorithm is evaluated. Three experiments are designed to study the behavior of different DE variants on a set of benchmark problems by using different performance measures proposed in the specialized literature. The first experiment analyzes the behavior of four DE variants in 24 test functions considering dimensionality and the type of constraints of the problem. The second experiment presents a more in-depth analysis on two DE variants by varying two parameters (the scale factor F and the population size NP), which control the convergence of the algorithm. From the results obtained, a simple but competitive combination of two DE variants is proposed and compared against state-of-the-art DE-based algorithms for constrained optimization in the third experiment. The study in this paper shows (1) important information about the behavior of DE in constrained search spaces and (2) the role of this knowledge in the correct combination of variants, based on their capabilities, to generate simple but competitive approaches.     

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.     

求解复杂约束优化问题的多策略混合麻雀搜索算法

针对麻雀搜索算法面对具有强约束、非凸性和不可微特征的复杂问题所存在的开发与探索能力不平衡、易陷入局部最优、过早收敛和种群多样性较低等不足,提出一种求解复杂约束优化问题的多策略混合麻雀搜索算法.首先,利用反向学习策略构建双向初始化机制,以达到获得分布更优的初始种群的目的;其次,设计一种基于交叉与变异算子的位置更新公式,扩大搜索范围,丰富搜索机制,以平衡算法探索和开发能力,同时提高算法的收敛精度和速度;最后,使用社区学习策略对种群进行精炼,强化开发能力与跳出局部极值的能力,并保持种群的多样性.分别在CEC2017的28个实数约束优化问题和1个工程优化问题上进行了性能评估,实验结果表明,所提出的算法对比其他优化算法具有寻优能力强、收敛精度高、收敛速度快等优势,可有效解决复杂约束优化问题.     

Enhanced opposition-based differential evolution for solving high-dimensional continuous optimization problems

This paper presents a novel algorithm based on generalized opposition-based learning (GOBL) to improve the performance of differential evolution (DE) to solve high-dimensional optimization problems efficiently. The proposed approach, namely GODE, employs similar schemes of opposition-based DE (ODE) for opposition-based population initialization and generation jumping with GOBL. Experiments are conducted to verify the performance of GODE on 19 high-dimensional problems with D = 50, 100, 200, 500, 1,000. The results confirm that GODE outperforms classical DE, real-coded CHC (crossgenerational elitist selection, heterogeneous recombination, and cataclysmic mutation) and G-CMA-ES (restart covariant matrix evolutionary strategy) on the majority of test problems.     

差异演化算法及其改进形式的综述

差异演化算法是一种基于群体差异的演化算法,群体中每个向量代表问题的一个候选解,该算法利用向量之间的差异扰动整个种群,求解问题的最优解.综述了差异演化算法的基本原理、常用测试函数和算法优缺点,讨论了多种改进手段,如:三角变异、混沌理论、逆向计算、均匀设计表等,并给出了未来可能的研究方向.     

A new improved whale optimization algorithm with joint search mechanisms for high-dimensional global optimization problems

Similar to other swarm-based algorithms, the recently developed whale optimization algorithm (WOA) has the problems of low accuracy and slow convergence. It is also easy to fall into local optimum. Moreover, WOA and its variants cannot perform well enough in solving high-dimensional optimization problems. This paper puts forward a new improved WOA with joint search mechanisms called JSWOA for solving the above disadvantages. First, the improved algorithm uses tent chaotic map to maintain the diversity of the initial population for global search. Second, a new adaptive inertia weight is given to improve the convergence accuracy and speed, together with jump out from local optimum. Finally, to enhance the quality and diversity of the whale population, as well as increase the probability of obtaining global optimal solution, opposition-based learning mechanism is used to update the individuals of the whale population continuously during each iteration process. The performance of the proposed JSWOA is tested by twenty-three benchmark functions of various types and dimensions. Then, the results are compared with the basic WOA, several variants of WOA and other swarm-based intelligent algorithms. The experimental results show that the proposed JSWOA algorithm with multi-mechanisms is superior to WOA and the other state-of-the-art algorithms in the competition, exhibiting remarkable advantages in the solution accuracy and convergence speed. It is also suitable for dealing with high-dimensional global optimization problems.

     

一种双种群差分蜂群算法

人工蜂群算法(ABC)是一种基于蜜蜂群智能搜索行为的随机优化算法.为了有效改善人工蜂群算法的性能,结合差分进化算法,提出一种新的双种群差分蜂群算法(BDABC).该算法首先通过基于反向学习的策略初始化种群,使得初始化的个体尽可能均匀分布在搜索空间,然后将种群中的个体随机分成两组,每组采用不同的优化策略同时进行寻优,并通过在两群体之间引入交互学习的思想,来提高算法的收敛速度.基于6个标准测试函数的仿真实验表明,BDABC算法能有效避免早熟收敛,全局优化能力和收敛速率都有显著提高.     

A modified artificial bee colony algorithm

Artificial bee colony algorithm (ABC) is a relatively new optimization technique which has been shown to be competitive to other population-based algorithms. However, there is still an insufficiency in ABC regarding its solution search equation, which is good at exploration but poor at exploitation. Inspired by differential evolution (DE), we propose an improved solution search equation, which is based on that the bee searches only around the best solution of the previous iteration to improve the exploitation. Then, in order to make full use of and balance the exploration of the solution search equation of ABC and the exploitation of the proposed solution search equation, we introduce a selective probability P and get the new search mechanism. In addition, to enhance the global convergence, when producing the initial population, both chaotic systems and opposition-based learning methods are employed. The new search mechanism together with the proposed initialization makes up the modified ABC (MABC for short), which excludes the probabilistic selection scheme and scout bee phase. Experiments are conducted on a set of 28 benchmark functions. The results demonstrate good performance of MABC in solving complex numerical optimization problems when compared with two ABC-based algorithms.