Modified Immune Clonal Constrained Multi-Objective Optimization Algorithm

针对约束多目标优化问题,提出修正免疫克隆约束多目标优化算法.该算法通过引进一个约束处理策略,用一个修正算法对个体的目标函数值进行修正,并对修正后的目标函数值采用免疫克隆算法进行优化,用一个精英种群对可行非支配解进行存储.该算法在优化过程中,既保留了非支配可行解,也充分利用了约束偏离值小的非可行解,同时引进整体克隆策略来提高解分布的多样性.通过对约束多目标问题的各项性能指标的测试以及和对比算法的比较可以看出:该算法在处理约束多目标优化测试问题时,所得解的多样性得到了一定的提高.同时,解的收敛性和均匀性也得到了一定的改进.     

基于局部搜索与混合多样性策略的多目标粒子群算法

为了提高算法的收敛性与非支配解集的多样性,提出一种基于局部搜索与混合多样性策略的多目标粒子群算法（LH-MOPSO）.该算法使用增广Lagrange乘子法对非支配解进行局部搜索以快速接近Pareto最优解;利用基于改进的Maximin适应值函数与拥挤距离的混合多样性策略对非支配解集进行维护以保留解的多样性,同时引入高斯变异算子以避免算法早熟收敛;最后针对多目标约束优化问题,给出一种有效的约束处理方法.实验研究表明该算法具有良好的优化性能.     

用于约束多目标优化问题的混合粒子群算法

针对约束多目标优化问题,结合Pareto支配思想、锦标赛选择和排挤距离技术,采用双种群搜索策略,引进免疫机制,对传统的粒子更新策略进行改进,提出一种用于求解约束多目标优化问题的混合粒子群算法。通过4个标准约束多目标函数进行测试,测试结果表明,该方法有效可行,相比传统多目标优化算法更优。     

免疫克隆多目标优化算法求解约束优化问题

针对现有的约束处理技术的一些不足之处,提出一种用于求解约束优化问题的算法——免疫克隆多目标优化算法(immune clonal multi-objective optimization algorithm,简称ICMOA).算法的主要特点是通过将约束条件转化为一个目标,从而将问题转化为两个目标的多目标优化问题.引入多目标优化中的Pareto-支配的概念,每一个个体根据其被支配的程度进行克隆、变异及选择等操作.克隆操作实现了全局择优,有利于得到高质量的解;变异操作提高算法的局部搜索能力,有利于所得解的多样性;选择操作有利于算法向着最优搜索,而且加快了收敛速度.基于抗体群的随机状态转移过程,证明该算法具有全局收敛性.通过对13个标准测试问题的测试,并与已有算法进行比较。结果表明,该算法在收敛速度和求解精度上均具有一定的优势.     

免疫克隆算法求解动态多目标优化问题

求解动态多目标优化(dynamic multi-objective optimization,简称DMO)问题的主要困难在于目标函数、约束条件或者相关的问题参数是随时间不断变化的.基于免疫克隆选择学说,提出一种用于解决DMO问题的新算法--动态多目标免疫克隆优化(immune clonal algorithm for DMO,简称ICADMO).该算法改进了现有的克隆策略,采用整体克隆的方式;在选择策略上,根据Pareto-占优的概念,将抗体群中的个体分为支配个体和非支配个体,对非支配个体进行选择.采用3个特色算子,使其很好地保持了所得解的多样性、均匀性和收敛性.通过数值实验,与DBM(direction-based method)算法进行比较,结果表明,新算法在收敛性、多样性以及解分布的广度方面都体现了很好的性能.     

一种改进的基于约束支配的多目标进化算法

针对传统优化方法在处理带约束的多目标优化问题上的不足进行了分析，将多目标进化算法以及约束支配的概念结合起来，重新定义了种群个体间的支配关系，避免了罚函数法因惩罚系数不合适而出现优化结果为非可行解的情况。并且结合惩罚值改进了选择算子和适应值分配机制，避免出现早熟收敛。同时，采用精共策略，让精英个体参与遗传操作，加快算法收敛速度。通过算例分析可知，将多目标进化算法以及约束支配的概念应用到浮筒配置优化方案是可行的、有效的。     

多目标扰动生物地理学优化算法

提出一种多目标扰动生物地理学优化算法(MDBBO) 来求解多目标优化问题(MOPs). 该算法基于现有群体中非支配可行解的比率, 联合个体非支配等级排序和拥挤距离对个体进行评价; 在生物地理迁移策略基础上提出扰动迁移算子并应用于群体进化, 增强群体多样性; 应用归档种群来保存所获得的非支配可行解, 并用循环拥挤距离法对其更新, 确保群体的均匀分布性. 通过标准函数测试以及与经典算法比较表明了该算法求解MOPs 的有效性.     

多目标扰动生物地理学优化算法

提出一种多目标扰动生物地理学优化算法(MDBBO)来求解多目标优化问题(MOPs).该算法基于现有群体中非支配可行解的比率,联合个体非支配等级排序和拥挤距离对个体进行评价;在生物地理迁移策略基础上提出扰动迁移算子并应用于群体进化,增强群体多样性;应用归档种群来保存所获得的非支配可行解,并用循环拥挤距离法对其更新,确保群体的均匀分布性.通过标准函数测试以及与经典算法比较表明了该算法求解MOPs的有效性.     

基于约束支配多目标进化算法的航路改航方法

针对具有多目标多约束特征的航路改航问题,提出了基于约束支配的非支配排序多目标进化算法(NSGA-Ⅱ)的路径规划方法.采用了约束支配规则处理空中交通管制规则中关于改航的多个约束条件,建立个体解之间的约束支配关系.在产生初始群体时进行分区产生随机的浮点坐标,并提前筛选出符合航向角改变量约束的可接受解,增加删除算子用以处理不可接受的不可行解,进化算法迭代完成后,建立优化算子用以优化迭代产生的路径.在块状和离散状飞行限制区条件下分别进行仿真实验,实验结果表明,该方法能产生符合目标函数优化且路径复杂度较小的最优解.     

求解约束高维多目标问题的分解约束支配NSGA-II优化算法

针对多目标进化算法处理约束高维多目标优化问题时出现解的分布性和收敛性差、易陷入局部最优解问题,采用Pareto支配、分解与约束支配融合的方法,提出一种基于分解约束支配NSGA-II优化算法(DBCDP-NSGA-II).该算法在保留NSGA-II中快速非支配排序的基础上,首先采用Pareto支配对种群进行支配排序;然后根据解的性质采用分解约束支配(DBCDP)惩罚等价解,保留稀疏区域的可行解和非可行解,提高种群的分布性、多样性和收敛性;最后采用个体到权重向量的垂直距离和拥挤度距离对临界值进行再排序,直到选出N个最优个体进入下一次迭代.以约束DTLZ问题中C-DTLZ1、C-DTLZ2、DTLZ8、DTLZ9测试函数为例,将所提出的算法与C-NSGA-II、C-NSGA-III、C-MOEA/D和C-MOEA/DD进行对比分析.仿真结果表明,DBCDP-NSGA-II所得最优解分布更加均匀,具有更好的全局收敛性.     

A modified objective function method with feasible-guiding strategy to solve constrained multi-objective optimization problems

For constrained multi-objective optimization problems (CMOPs), how to preserve infeasible individuals and make use of them is a problem to be solved. In this case, a modified objective function method with feasible-guiding strategy on the basis of NSGA-II is proposed to handle CMOPs in this paper. The main idea of proposed algorithm is to modify the objective function values of an individual with its constraint violation values and true objective function values, of which a feasibility ratio fed back from current population is used to keep the balance, and then the feasible-guiding strategy is adopted to make use of preserved infeasible individuals. In this way, non-dominated solutions, obtained from proposed algorithm, show superiority on convergence and diversity of distribution, which can be confirmed by the comparison experiment results with other two CMOEAs on commonly used constrained test problems.     

改进的约束优化多目标遗传算法及工程应用

利用多目标法处理约束条件,提出一种改进的基于多目标优化的遗传算法用于求解约束优化问题。该算法将约束优化问题转化为两个目标的多目标优化问题; 利用庄家法构造非劣个体,将种群分为支配子种群和非支配子种群,以一定概率分别从支配子种群和非支配子种群中选择个体进行算术交叉操作,引导个体逐步向极值点靠近,增强算法的局部搜索能力,对非支配子种群进行多样性变异操作。8个标准测试函数和3个工程应用的仿真实验结果表明了该算法的有效性。     

A surrogate-assisted evolution strategy for constrained multi-objective optimization

In many real-world optimization problems, several conflicting objectives must be achieved and optimized simultaneously and the solutions are often required to satisfy certain restrictions or constraints. Moreover, in some applications, the numerical values of the objectives and constraints are obtained from computationally expensive simulations. Many multi-objective optimization algorithms for continuous optimization have been proposed in the literature and some have been incorporated or used in conjunction with expert and intelligent systems. However, relatively few of these multi-objective algorithms handle constraints, and even fewer, use surrogates to approximate the objective or constraint functions when these functions are computationally expensive. This paper proposes a surrogate-assisted evolution strategy (ES) that can be used for constrained multi-objective optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. Such an algorithm can be incorporated into an intelligent system that finds approximate Pareto optimal solutions to simulation-based constrained multi-objective optimization problems in various applications including engineering design optimization, production management and manufacturing. The main idea in the proposed algorithm is to generate a large number of trial offspring in each generation and use the surrogates to predict the objective and constraint function values of these trial offspring. Then the algorithm performs an approximate non-dominated sort of the trial offspring based on the predicted objective and constraint function values, and then it selects the most promising offspring (those with the smallest predicted ranks from the non-dominated sort) to become the actual offspring for the current generation that will be evaluated using the expensive objective and constraint functions. The proposed method is implemented using cubic radial basis function (RBF) surrogate models to assist the ES. The resulting RBF-assisted ES is compared with the original ES and to NSGA-II on 20 test problems involving 2–15 decision variables, 2–5 objectives and up to 13 inequality constraints. These problems include well-known benchmark problems and application problems in manufacturing and robotics. The numerical results showed that the RBF-assisted ES generally outperformed the original ES and NSGA-II on the problems used when the computational budget is relatively limited. These results suggest that the proposed surrogate-assisted ES is promising for computationally expensive constrained multi-objective optimization.     

A multi-objective differential evolutionary algorithm for constrained multi-objective optimization problems with low feasible ratio

Most current evolutionary multi-objective optimization (EMO) algorithms perform well on multi-objective optimization problems without constraints, but they encounter difficulties in their ability for constrained multi-objective optimization problems (CMOPs) with low feasible ratio. To tackle this problem, this paper proposes a multi-objective differential evolutionary algorithm named MODE-SaE based on an improved epsilon constraint-handling method. Firstly, MODE-SaE self-adaptively adjusts the epsilon level in line with the maximum and minimum constraint violation values of infeasible individuals. It can prevent epsilon level setting from being unreasonable. Then, the feasible solutions are saved to the external archive and take part in the population evolution by a co-evolution strategy. Finally, MODE-SaE switches the global search and local search by self-switching parameters of search engine to balance the convergence and distribution. With the aim of evaluating the performance of MODE-SaE, a real-world problem with low feasible ratio in decision space and fourteen bench-mark test problems, are used to test MODE-SaE and five other state-of-the-art constrained multi-objective evolution algorithms. The experimental results fully demonstrate the superiority of MODE-SaE on all mentioned test problems, which indicates the effectiveness of the proposed algorithm for CMOPs which have low feasible ratio in search space.     

多目标自适应和声搜索算法

提出了一种利用Pareto支配来求解多目标优化问题的自适应和声搜索算法（MOSAHS）。该算法利用外部种群来保存非支配解,为了保持非支配解的多样性,提出了一种基于拥挤度的删除策略,这个策略能较好地度量个体的拥挤程度。用5个标准测试函数对其进行测试,并与其他多目标优化算法相比较。实验结果表明,与其他的算法相比,提出的算法在逼近性和均匀性两方面都有很好的表现,是一种有效的多目标和声搜索算法。     

多模态多目标差分进化算法求解非线性方程组

针对当前算法在求解非线性方程组时面临解的个数不完整、精确度不高、收敛速度慢等问题进行了研究,提出一种多模态多目标差分进化算法。首先将非线性方程组转换为多模态多目标优化问题,初始化一个随机种群并对种群中全部个体进行评价;然后通过非支配解排序和决策空间拥挤距离选择机制,挑选种群中的一半优质个体进行变异;接着在变异过程中采用一种新的变异策略和边界处理方法以增加解的多样性;最后通过交叉和选择机制使优质个体进行进化,直到搜索到全部最优解。在所选测试函数集和工程实例上的实验结果表明,该算法能有效地搜索到非线性方程组的解,并通过与当前四个算法进行比较,该算法在解的数量和成功率上具有优越性。     

An evolutionary algorithm with directed weights for constrained multi-objective optimization

When solving constrained multi-objective optimization problems (CMOPs), keeping infeasible individuals with good objective values and small constraint violations in the population can improve the performance of the algorithms, since they provide the information about the optimal direction towards Pareto front. By taking the constraint violation as an objective, we propose a novel constraint-handling technique based on directed weights to deal with CMOPs. This paper adopts two types of weights, i.e. feasible and infeasible weights distributing on feasible and infeasible regions respectively, to guide the search to the promising region. To utilize the useful information contained in infeasible individuals, this paper uses infeasible weights to maintain a number of well-diversified infeasible individuals. Meanwhile, they are dynamically changed along with the evolution to prefer infeasible individuals with better objective values and smaller constraint violations. Furthermore, 18 test instances and 2 engineering design problems are used to evaluate the effectiveness of the proposed algorithm. Several numerical experiments indicate that the proposed algorithm outperforms four compared algorithms in terms of finding a set of well-distributed non-domination solutions.     

Vehicle power train optimization using multi-objective bird swarm algorithm

In this paper, a multi-objective bird swarm algorithm (MOBSA) is proposed to cope with multi-objective optimization problems. The algorithm is explored based on BSA which is an evolutionary algorithm suitable for single objective optimization. In this paper, non-dominated sorting approach is used to distinguish optimal solutions and parallel coordinates is applied to evaluate the distribution density of non-dominated solution and further update the external archive when it is full to overflowing, which ensure faster convergence and more widespread of Pareto front. Then, the MOBSA is adopted to optimize benchmark problems. The results demonstrate that MOBSA gets better performance compared with NSGA-II and MOPSO. Since a vehicle power train problem could be treated as a typical multi-objective optimization problem with constraints, with integration of constrained non-dominated solution, MOBSA is adopted to acquire optimal gear ratios and optimize vehicle power train. The results compared with other popular algorithm prove the proposed algorithm is more suitable for constrained multi-objective optimization problem in engineering field.