Cauchy Grey Wolf Optimiser for continuous optimisation problems

ABSTRACT

Grey Wolf Optimiser (GWO) is a recently developed optimisation approach to solve complex non-linear optimisation problems. It is relatively simple and leadership-hierarchy based approach in the class of Swarm Intelligence based algorithms. For solving complex real-world non-linear optimisation problems, the search equation provided in GWO is not of sufficient explorative behaviour. Therefore, in the present paper, an attempt has been made to increase the exploration capability along with the exploitation of a search space by proposing an improved version of classical GWO. The proposed algorithm is named as Cauchy-GWO. In Cauchy-GWO Cauchy operator has been integrated in which first two new wolves are generated with the help of Cauchy distributed random numbers and then another new wolf is generated by taking the convex combination of these new wolves. The performance of Cauchy-GWO is exhibited on standard IEEE CEC 2014 benchmark problem set. Statistical analysis of the results on CEC 2014 benchmark set and popular evaluation criteria, Performance Index (PI) proves that Cauchy-GWO outperforms GWO in terms of error values defined in IEEE CEC 2014 benchmarks collection. Later on in the paper, GWO and Cauchy-GWO algorithms have been used to solve three well-known engineering application problems and two problems of reliability. From the analysis conducted in the present paper, it can be concluded that the proposed algorithm, Cauchy-GWO is reliable and efficient algorithm to solve continuous benchmark test problems, as well as real-life applications problems.     

I-GWO and Ex-GWO: improved algorithms of the Grey Wolf Optimizer to solve global optimization problems

In this paper, two novel meta-heuristic algorithms are introduced to solve global optimization problems inspired by the Grey Wolf Optimizer (GWO) algorithm. In the GWO algorithm, wolves are likely to be located in regions close to each other. Therefore, as they catch the hunt (approaching the solution), they may create an intensity in the same or certain regions. In this case, the mechanism to prevent the escape of the hunt may not work well. First, the proposed algorithm is the expanded model of the GWO algorithm that is called expanded Grey Wolf Optimizer. In this method, the same as GWO, alpha, beta, and delta play the role of the main three wolves. However, the next wolves select and update their positions according to the previous and the first three wolves in each iteration. Another proposed algorithm is based on the incremental model and is, therefore, called incremental Grey Wolf Optimizer. In this method, each wolf updates its own position based on all the wolves selected before it. There is the possibility of finding solutions (hunts) quicker than according to other algorithms in the same category. However, they may not always guarantee to find a good solution because of their act dependent on each other. Both algorithms focus on exploration and exploitation. In this paper, the proposed algorithms are simulated over 33 benchmark functions and the related results are compared with well-known optimization algorithms. The results of the proposed algorithms seem to be good solutions for various problems.

     

Grey Wolf Optimizer

This work proposes a new meta-heuristic called Grey Wolf Optimizer (GWO) inspired by grey wolves (Canis lupus). The GWO algorithm mimics the leadership hierarchy and hunting mechanism of grey wolves in nature. Four types of grey wolves such as alpha, beta, delta, and omega are employed for simulating the leadership hierarchy. In addition, the three main steps of hunting, searching for prey, encircling prey, and attacking prey, are implemented. The algorithm is then benchmarked on 29 well-known test functions, and the results are verified by a comparative study with Particle Swarm Optimization (PSO), Gravitational Search Algorithm (GSA), Differential Evolution (DE), Evolutionary Programming (EP), and Evolution Strategy (ES). The results show that the GWO algorithm is able to provide very competitive results compared to these well-known meta-heuristics. The paper also considers solving three classical engineering design problems (tension/compression spring, welded beam, and pressure vessel designs) and presents a real application of the proposed method in the field of optical engineering. The results of the classical engineering design problems and real application prove that the proposed algorithm is applicable to challenging problems with unknown search spaces.     

强化狼群等级制度的灰狼优化算法

针对灰狼优化(Grey wolf optimization, GWO)算法在处理复杂优化问题时优化精度不高,易陷于局部最优等问题,提出了一种强化狼群等级制度的灰狼优化(GWO based on strengthening the hierarchy of wolves, GWOSH)算法。该算法为灰狼个体设置了跟随狩猎和自主探索两种狩猎模式,并根据自身等级情况来控制选择狼群的狩猎模式。在跟随狩猎模式中,灰狼个体以等级高于自身的灰狼的位置信息来指引自己到达最优解区域;而在自主探索模式中,灰狼个体会同时审视等级高于自身的灰狼的位置信息和自身位置信息,并基于这些信息自主判断猎物的位置,同时两种更新模式都将引入优胜劣汰选择规则来确保种群的狩猎方向。对12个基准测试函数进行优化的结果表明：与已有的算法相比,GWOSH算法的全局搜索能力更强,更能有效避免易早熟收敛的问题,更适用于求解高维的复杂优化问题。     

基于醉汉漫步和反向学习的灰狼优化算法

灰狼优化算法在优化后期易陷入局部最优,在求解高维函数时因其复杂度更高,陷入局部最优概率更大,针对上述问题提出基于醉汉漫步和反向学习的混合灰狼优化算法（DGWO）。在迭代过程中对每代种群中优势狼与最差狼进行反向学习并进行比较、重新排序后保留前3的狼,同时将采用醉汉漫步机制更新领导狼,参数A和C采用系数标量而不是GWO原始算法中的系数向量。通过10个标准测试函数(100维、500维和1 000维)以及10维的CEC2013测试函数验证了算法的性能,并与PSO、GWO-CS和GWO算法进行了比较, 结果表明,该混合灰狼优化算法 在精度和收敛速度上都具有优势。此外,将改进的灰狼优化算法应用于两级运算放大器参数设计,以开环低频增益最大化为目标,验证该算法的实用性。     

Multi-objective grey wolf optimizer: A novel algorithm for multi-criterion optimization

Due to the novelty of the Grey Wolf Optimizer (GWO), there is no study in the literature to design a multi-objective version of this algorithm. This paper proposes a Multi-Objective Grey Wolf Optimizer (MOGWO) in order to optimize problems with multiple objectives for the first time. A fixed-sized external archive is integrated to the GWO for saving and retrieving the Pareto optimal solutions. This archive is then employed to define the social hierarchy and simulate the hunting behavior of grey wolves in multi-objective search spaces. The proposed method is tested on 10 multi-objective benchmark problems and compared with two well-known meta-heuristics: Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D) and Multi-Objective Particle Swarm Optimization (MOPSO). The qualitative and quantitative results show that the proposed algorithm is able to provide very competitive results and outperforms other algorithms. Note that the source codes of MOGWO are publicly available at http://www.alimirjalili.com/GWO.html.     

混合苍狼优化算法在全局最优中的应用研究

在分析标准苍狼优化算法（GWO）的开发与探索性能基础上,提出了一种混合苍狼优化算法（MAR- GWO）,搜索域得到了全面的扩展,其中针对[α、][β、][δ]领导层苍狼,引入自主搜索行为来加大其优化力度与促进速度的提高,对性能较差搜索狼采取淘汰重组机制以提高搜索效率,又采取概率差分变异行为增加了个体多样性,从而避免局部最优。为了验证MAR-GWO算法有效性,对13个全局优化问题进行实验,分别与GWO、GWO-EPD（改进的苍狼优化算法）、PSO、EA等算法进行了对比测试,从实验结果来看,MAR-GWO算法寻优成功率相对较高、收敛速度快,不易陷入局部最优,在智能算法中具有很强的竞争力。     

An optimized grey wolf optimizer based on a mutation operator and eliminating-reconstructing mechanism and its application

Due to its simplicity and ease of use, the standard grey wolf optimizer (GWO) is attracting much attention. However, due to its imperfect search structure and possible risk of being trapped in local optima, its application has been limited. To perfect the performance of the algorithm, an optimized GWO is proposed based on a mutation operator and eliminating-reconstructing mechanism (MR-GWO). By analyzing GWO, it is found that it conducts search with only three leading wolves at the core, and balances the exploration and exploitation abilities by adjusting only the parameter a, which means the wolves lose some diversity to some extent. Therefore, a mutation operator is introduced to facilitate better searching wolves, and an eliminating- reconstructing mechanism is used for the poor search wolves, which not only effectively expands the stochastic search, but also accelerates its convergence, and these two operations complement each other well. To verify its validity, MR-GWO is applied to the global optimization experiment of 13 standard continuous functions and a radial basis function (RBF) network approximation experiment. Through a comparison with other algorithms, it is proven that MR-GWO has a strong advantage.     

Solution of non-convex economic load dispatch problem using Grey Wolf Optimizer

Grey Wolf Optimizer (GWO) is a recently developed meta-heuristic search algorithm inspired by grey wolves (Canis lupus), which simulate the social stratum and hunting mechanism of grey wolves in nature and based on three main steps of hunting: searching for prey, encircling prey and attacking prey. This paper presents the application of GWO algorithm for the solution of non-convex and dynamic economic load dispatch problem (ELDP) of electric power system. The performance of GWO is tested for ELDP of small-, medium- and large-scale power systems, and the results are verified by a comparative study with lambda iteration method, Particle Swarm Optimization algorithm, Genetic Algorithm, Biogeography-Based Optimization, Differential Evolution algorithm, pattern search algorithm, NN-EPSO, FEP, CEP, IFEP and MFEP. Comparative results show that the GWO algorithm is able to provide very competitive results compared to other well-known conventional, heuristics and meta-heuristics search algorithms.     

Multi-objective Grey Wolf Optimizer for improved cervix lesion classification

Cervical cancer is one of the vital and most frequent cancers, but can be cured effectively if diagnosed in the early stage. This is a novel effort towards effective characterization of cervix lesions from contrast enhanced CT-Scan images to provide a reliable and objective discrimination between benign and malignant lesions. Performance of such classification models mostly depends on features used to represent samples in a training dataset. Selection of optimal feature subset here is NP-hard; where, randomized algorithms do better. In this paper, Grey Wolf Optimizer (GWO), which is a population based meta-heuristic inspired by the leadership hierarchy and hunting mechanism of grey wolves has been utilized for feature selection. The traditional GWO is applicable for continuous single objective optimization problems. Since, feature selection is inherently multi-objective; this paper proposes two different approaches for multi-objective binary GWO algorithms. One is a scalarized approach to multi-objective GWO (MOGWO) and the other is a Non-dominated Sorting based GWO (NSGWO). These are used for wrapper based feature selection that selects optimal textural feature subset for improved classification of cervix lesions. For experiments, contrast enhanced CT-Scan (CECT) images of 62 patients have been used, where all lesions had been recommended for surgical biopsy by specialist. Gray-level co-occurrence matrix based texture features are extracted from two-level decomposition of wavelet coefficients of cervix regions extracted from CECT images. The results of proposed approaches are compared with mostly used meta-heuristics such as genetic algorithm (GA) and firefly algorithm (FA) for multi-objective optimization. With better diversification and intensification, GWO obtains Pareto solutions, which dominate the solutions obtained by GA and FA when assessed on the utilized cervix lesion cases. Cervix lesions are up to 91% accurately classified as benign and malignant with only five features selected by NSGWO. A two-tailed t-test was conducted by hypothesizing the mean F-score obtained by the proposed NSGWO method at significance level = 0.05. This confirms that NSGWO performs significantly better than other methods for the real cervix lesion dataset in hand. Further experiments were conducted on high dimensional microarray gene expression datasets collected online. The results demonstrate that the proposed method performs significantly better than other methods selecting relevant genes for high-dimensional, multi-category cancer diagnosis with an average of 12.82% improvement in F-score value.     

基于翻筋斗觅食策略的灰狼优化算法

灰狼优化算法(grey wolf optimization,GWO)存在收敛的不合理性等缺陷,目前对GWO算法的收敛性改进方式较少,除此之外,当GWO迭代至后期,所有灰狼个体都逼近α狼、β狼、δ狼,导致算法陷入局部最优。针对以上问题,提出了一种增强型的灰狼优化算法(disturbance and somersault foraging-grey wolf optimization,DSF-GWO)。首先引入一种扰动因子,平衡了算法的开采和勘探能力;其次引入翻筋斗觅食策略,在后期使其不陷入局部最优的同时也使得前期的群体多样性略有提升。对DSF-GWO算法的寻优性能进行验证,选取14个单/多峰目标函数进行实验,在相同的参数设置下,结果表明DSF-GWO算法在寻优性能上较GWO算法有明显优势。     

协调探索和开发能力的改进灰狼优化算法

提出一种协调探索和开发能力的灰狼优化算法.利用佳点集方法初始化灰狼个体的位置,为全局搜索多样性奠定基础;为协调算法的全局探索和局部开发能力,给出一种基于正切三角函数描述的非线性动态变化控制参数;为加快算法的收敛速度,受粒子群优化算法个体记忆功能的启发,设计一种新的个体位置更新公式.10个标准函数的测试结果表明,改进灰狼优化(IGWO)算法能够有效地协调其对问题搜索空间的探索和开发能力.     

Multi-strategy ensemble grey wolf optimizer and its application to feature selection

To overcome the limitation of single search strategy of grey wolf optimizer (GWO) in solving various function optimization problems, we propose a multi-strategy ensemble GWO (MEGWO) in this paper. The proposed MEGWO incorporates three different search strategies to update the solutions. Firstly, the enhanced global-best lead strategy can improve the local search ability of GWO by fully exploiting the search space around the current best solution. Secondly, the adaptable cooperative strategy embeds one-dimensional update operation into the framework of GWO to provide a higher population diversity and promote the global search ability. Thirdly, the disperse foraging strategy forces a part of search agents to explore a promising area based on a self-adjusting parameter, which contributes to the balance between the exploitation and exploration. We conducted numerical experiments based on various functions form CEC2014. The obtained results are compared with other three modified GWO and seven state-of-the-art algorithms. Furthermore, feature selection is employed to investigate the effectiveness of MEGWO on real-world applications. The experimental results show that the proposed algorithm which integrate multiple improved search strategies, outperforms other variants of GWO and other algorithms in terms of accuracy and convergence speed. It is validated that MEGWO is an efficient and reliable algorithm not only for optimization of functions with different characteristics but also for real-world optimization problems.     

基于混沌灰狼优化算法的SVM分类器研究

支持向量机(SVM)是在分类问题下建立的一个运算小型数据集,可实现非线性高纬度分类,有很好的扩展能力。但是,在传统SVM的训练过程中,SVM运算结果的好坏与参数选择关系密切,而且目前使用的参数选择算法有很多缺陷。因此,针对上述问题,在灰狼算法（GWO）中加入混沌序列,改变狼群初始分布规律,构建混沌灰狼优化算法(CGWO),增强狼群分布均匀性以及狼群查找遍历性,极大提高GWO算法的运算速度和运算准确性,最终更好地优化SVM。使用Mirjalili提供的开源数据与原有数据混合作为向量机的测试集进行实验对比分析,实验结果表明,CGWO算法具有明显的性能提高;用混沌灰狼算法优化的 SVM和灰狼优化算法SVM、人工蜂群SVM、万有引力搜索SVM以及传统算法优化的 SVM相比,其运算准确率更高、误差更低、花费时间更少。     

An efficient modified grey wolf optimizer with Lévy flight for optimization tasks

The grey wolf optimizer (GWO) is a new efficient population-based optimizer. The GWO algorithm can reveal an efficient performance compared to other well-established optimizers. However, because of the insufficient diversity of wolves in some cases, a problem of concern is that the GWO can still be prone to stagnation at local optima. In this article, an improved modified GWO algorithm is proposed for solving either global or real-world optimization problems. In order to boost the efficacy of GWO, Lévy flight (LF) and greedy selection strategies are integrated with the modified hunting phases. LF is a class of scale-free walks with randomly-oriented steps according to the Lévy distribution. In order to investigate the effectiveness of the modified Lévy-embedded GWO (LGWO), it was compared with several state-of-the-art optimizers on 29 unconstrained test beds. Furthermore, 30 artificial and 14 real-world problems from CEC2014 and CEC2011 were employed to evaluate the LGWO algorithm. Also, statistical tests were employed to investigate the significance of the results. Experimental results and statistical tests demonstrate that the performance of LGWO is significantly better than GWO and other analyzed optimizers.     

Ludo game-based metaheuristics for global and engineering optimization

This paper proposes a Ludo game-based strategy to enhance the ability of swarm algorithms to solve numerous global optimization problems. The proposed strategy simulates the rules of playing the game Ludo using two or four players to perform the update process for different swarm intelligent behaviors. The proposed approach is named the Ludo Game-based Swarm Intelligence (LGSI) Algorithm. The LGSI algorithm uses the concepts of two and four players to enhance the exploration and exploitation of the optimization methods. In the proposed LGSI, a player is represented by a swarm algorithm, for example, in the two-player concept; Moth Flame Optimization (MFO) and the Grasshopper Optimization Algorithm (GOA) are selected, while in the four-player version, two other algorithms, the Sine Cosine Algorithm (SCA) and Gray Wolf Optimization (GWO), are added. In the proposed LGSI algorithm, the functional behaviors of all the used algorithms are different; also, there is no similarity among algorithmic behaviors except for convergence towards the global optimum, which is a common interest for all. However, the other algorithms share the same platform with this strategy, so in this case, competitive behavior may not be underestimated. The proposed LGSI algorithm shares positions among all the algorithms used during the search for the optimal solution. The performance of the LGSI algorithm is tested on a set of CEC2005 benchmark problems and engineering problems and is compared with the original versions of the utilized algorithms and a variety of other state-of-the-art algorithms. The experimental results show that the LGSI algorithm can provide promising and competitive results.     

求解约束优化问题的改进灰狼优化算法

针对基本灰狼优化(GWO)算法存在求解精度低、收敛速度慢、局部搜索能力差的问题,提出一种改进灰狼优化(IGWO)算法用于求解约束优化问题。该算法采用非固定多段映射罚函数法处理约束条件,将原约束优化问题转化为无约束优化问题,然后利用IGWO算法对转换后的无约束优化问题进行求解。在IGWO算法中,引入佳点集理论生成初始种群,为算法全局搜索奠定基础;为了提高局部搜索能力和加快收敛,对当前最优灰狼个体执行Powell局部搜索。采用几个标准约束优化测试问题进行仿真实验,结果表明该算法不仅克服了基本GWO的缺点,而且性能优于差分进化和粒子群优化算法。     

改进的灰狼优化算法及其高维函数和FCM优化

灰狼优化算法(GWO)具有较强的局部搜索能力和较快的收敛速度,但在解决高维和复杂的优化问题时存在全局搜索能力不足的问题.对此,提出一种改进的GWO,即新型反向学习和差分变异的GWO(ODGWO).首先,提出一种最优最差反向学习策略和一种动态随机差分变异算子,并将它们融入GWO中,以便增强全局搜索能力;然后,为了很好地平衡探索与开采能力以提升整体的优化性能,对算法前、后半搜索阶段分别采用单维操作和全维操作形成ODGWO;最后,将ODGWO用于高维函数和模糊C均值(FCM)聚类优化.实验结果表明,在许多高维Benchmark函数(30维、50维和1000维)优化上,ODGWO的搜索能力大幅度领先于GWO,与state-of-the-art优化算法相比,ODGWO具有更好的优化性能.在7个标准数据集的FCM聚类优化上, 与GWO、GWOepd和LGWO相比,ODGWO表现出了更好的聚类优化性能,可应用在更多的实际优化问题上.     

An improved firefly algorithm for solving dynamic multidimensional knapsack problems

There is a wide range of publications reported in the literature, considering optimization problems where the entire problem related data remains stationary throughout optimization. However, most of the real-life problems have indeed a dynamic nature arising from the uncertainty of future events. Optimization in dynamic environments is a relatively new and hot research area and has attracted notable attention of the researchers in the past decade. Firefly Algorithm (FA), Genetic Algorithm (GA) and Differential Evolution (DE) have been widely used for static optimization problems, but the applications of those algorithms in dynamic environments are relatively lacking. In the present study, an effective FA introducing diversity with partial random restarts and with an adaptive move procedure is developed and proposed for solving dynamic multidimensional knapsack problems. To the best of our knowledge this paper constitutes the first study on the performance of FA on a dynamic combinatorial problem. In order to evaluate the performance of the proposed algorithm the same problem is also modeled and solved by GA, DE and original FA. Based on the computational results and convergence capabilities we concluded that improved FA is a very powerful algorithm for solving the multidimensional knapsack problems for both static and dynamic environments.     

基于透镜成像学习策略的灰狼优化算法

在灰狼优化算法中, $ {{\pmb C}} $是一个重要的参数, 其功能是负责算法的勘探能力.目前, 针对参数$ {{\pmb C}} $的研究工作相对较少.另外, 在算法进化过程中, 群体中其他个体均向$\alpha$、$\beta$和$\delta$所在区域靠近以加快收敛速度.然而, 算法易陷入局部最优.为解决以上问题, 本文提出一种改进的灰狼优化算法(Lens imaging learning grey wolf optimizer algorithm, LIL-GWO).该算法首先分析了参数$ {{\pmb C}} $的作用, 提出一种新的参数$\pmb C$策略以平衡算法的勘探和开采能力; 同时, 分析了灰狼优化算法后期个体均向决策层区域聚集, 从而导致群体多样性较差, 提出一种基于光学透镜成像原理的反向学习策略以避免算法陷入局部最优.对LIL-GWO算法的收敛性进行了证明.选取12个通用的标准测试函数和30个CEC 2014测试函数进行实验, 在相同的适应度函数评价次数条件下, LIL-GWO算法在总体性能上优于基本GWO算法、改进GWO算法和其他比较算法.最后, 将LIL-GWO算法应用于辨识光伏模型的参数, 获得了满意的结果.