An orthogonal multi-objective evolutionary algorithm for multi-objective optimization problems with constraints

In this paper, an orthogonal multi-objective evolutionary algorithm (OMOEA) is proposed for multi-objective optimization problems (MOPs) with constraints. Firstly, these constraints are taken into account when determining Pareto dominance. As a result, a strict partial-ordered relation is obtained, and feasibility is not considered later in the selection process. Then, the orthogonal design and the statistical optimal method are generalized to MOPs, and a new type of multi-objective evolutionary algorithm (MOEA) is constructed. In this framework, an original niche evolves first, and splits into a group of sub-niches. Then every sub-niche repeats the above process. Due to the uniformity of the search, the optimality of the statistics, and the exponential increase of the splitting frequency of the niches, OMOEA uses a deterministic search without blindness or stochasticity. It can soon yield a large set of solutions which converges to the Pareto-optimal set with high precision and uniform distribution. We take six test problems designed by Deb, Zitzler et al., and an engineering problem (W) with constraints provided by Ray et al. to test the new technique. The numerical experiments show that our algorithm is superior to other MOGAS and MOEAs, such as FFGA, NSGAII, SPEA2, and so on, in terms of the precision, quantity and distribution of solutions. Notably, for the engineering problem W, it finds the Pareto-optimal set, which was previously unknown.     

自适应进化多目标粒子群优化算法

提出一种自适应进化粒子群优化算法以求解多目标优化问题．采用非支配排序策略和动态加权法选择最优粒子,引导种群飞行,提高Ｐａｒｅｔｏ解的多样性．采用动态惯性权重,提高其全局寻优能力．当种群的寻优能力减弱时,采用变异操作以引导粒子群跳出局部最优．通过ＺＤＴ１～ＺＤＴ４ 基准函数验证,该算法能够在保持优化解多样性的同时实现较好的收敛性．与其他多目标进化算法和多目标粒子群优化算法相比,该算法具有较好的性能．     

用多目标演化优化算法解决约束选址问题

约束选址问题是一个多目标约束优化问题，传统算法（加权法）一次只能得到一个候选解，用多目标演化优化算法对其进行求解，可以一次得到多个候选解，给决策者提供更多的选择余地，以期获得更大的利益，数字试验表明，该方法优于传统多目标优化方法。     

一种新的多目标改进和声搜索优化算法

针对标准和声搜索算法存在收敛不稳定及不能用于多目标优化问题的缺陷,通过引入交叉算子、自适应记忆内搜索概率和调节概率,改进了传统的和声搜索算法;根据Pareto支配关系,结合算法和声记忆库内信息完全共享的特性,提出了基于动态Pareto最优前沿的能够求解多目标优化问题的多目标改进和声搜索算法。通过几个典型函数的仿真测试表明,提出的算法能够高效稳定地收敛于Pareto最优前沿,获得分布均匀的Pareto解集。     

融合分区和局部搜索的多模态多目标优化

为解决多模态多目标优化中种群多样性维持难和所得等价解数量不足问题,基于分区搜索和局部搜索,本研究提出一种融合分区和局部搜索的多模态多目标粒子群算法(multimodal multi-objective particle swarm optimization combing zoning search and local search,ZLS-SMPSO-MM)。在所提算法中,整个搜索空间被分割成多个子空间以维持种群多样性和降低搜索难度;然后,使用已有的自组织多模态多目标粒子群算法在每个子空间搜索等价解和挖掘邻域信息,并利用局部搜索能力较强的协方差矩阵自适应算法对有潜力的区域进行精细搜索。通过14个多模态多目标优化问题测试,并与其他5种知名算法进行比较;实验结果表明ZLS-SMPSO-MM在决策空间能够找到更多的等价解,且整体性能要好于所比较算法。     

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.     

Medical image reconstruction using a multi-objective genetic local search algorithm

Image reconstruction from projections is a key problem in medical image analysis. In this paper, we cast image reconstruction from projections as a multi-objective problem. It is essential to choose some proper objective functions of the problem. We choose the square error, smoothness of the reconstructed image, and the maximum entropy as our objective functions of the problem. Then we introduce a hybrid algorithm comprising of multi-objective genetic and local search algorithms to reconstruct the image. Our algorithm has remarkable global performance. Our experiments show that we can get different results when we give different weights to different objective functions. We can also control the noise by giving different weights on different objective function. At the same time, we can adjust the parameter to let it have good local performance. Though the computation demands of the hybrid algorithm tends to be larger because of the random search of the GA, it is really a common feature of the global optimization method. Our results show that the hybrid algorithm is a more effective than the conventional method. We think our method is very promising for the medical imaging field.     

An improved model-based evolutionary algorithm for multi-objective optimization

The basic idea in the estimation of distribution algorithms is the replacement of heuristic operators with machine learning models such as regression models, clustering models, or classification models. So, recently, the model-based evolutionary algorithms (MBEAs) have been suggested in three groups: The estimation of distribution algorithms (EDAs), surrogate assisted evolutionary algorithms, and the inversed models to map from the objective space to the decision space. In this article, a new approach, based on an inversed model of Gaussian process and random forest framework, is proposed. The main idea is applying the process of random forest variable importance with a random grouping that determines some of the best assignment of decision variables to objective functions in order to form a Gaussian process in inverse models that maps to decision space the rich solutions which are discovered from objective space. Then these inverse models through sampling the objective space generate offspring. The proposed algorithm has been tested on the benchmark test suite for evolutionary algorithms (modified Deb K, Thiele L, Laumanns M, Zitzler E (DTLZ), and Walking Fish Group (WFG)) and indicates that the proposed method is a competitive and promising approach.     

基于多目标优化的联邦学习进化算法

传统联邦学习存在通信成本高、结构异构、隐私保护力度不足的问题,为此提出了一种联邦学习进化算法,应用稀疏进化训练算法降低通信成本,结合本地化差分隐私保护参与方隐私,同时采用NSGA-Ⅲ算法优化联邦学习全局模型的网络结构、稀疏性,调整数据可用性与隐私保护之间的关系,实现联邦学习全局模型有效性、通信成本和隐私性的均衡。不稳定通信环境下的实验结果表明,在MNIST和CIFAR-10数据集上,与FNSGA-Ⅲ算法错误率最低的解相比,该算法所得解的通信效率分别提高57.19%和52.17%,并且参与方实现了（3.46,10-4）和（6.52,10-4）-本地化差分隐私。在不严重影响全局模型准确率的前提下,该算法有效降低了联邦学习的通信成本并保护了参与方隐私。     

混合分散搜索的进化多目标优化算法

在进化多目标优化算法中,种群的多样性、对目标空间的搜索能力及算法的鲁棒性直接影响算法的收敛能力和解集的分散性。针对这些问题,提出了一种混合分散搜索的进化多目标优化算法(SSMOEA)。SSMOEA在混合分散搜索算法架构的同时,重新设计其多样性的选取策略,并引入协同进化机制。此外,为了提高算法的自适应性和鲁棒性,采用了一种新颖的自适应多交叉算子选择方法。SSMOEA与经典的多目标进化算法SPEA2、NSGA-Ⅱ和MOEA/D在12个基准测试函数上的对比结果表明,SSMOEA不仅在求得的Pareto最优解集的宽广性、均匀性和逼近性上有明显优势,而且算法的鲁棒性也有明显的提高。     

基于距离收敛量和历史信息密度的多目标进化算法

在筛选个体的过程中,多目标进化算法大都利用非支配信息和密度信息评价个体。但当个体互为非支配关系时,上述信息就难以区分个体的优劣从而影响算法性能。为了改善上述情况,本文提出了一种基于距离收敛量和历史信息密度的多目标进化算法。距离收敛量可以在非支配信息不能区分个体时评价个体的收敛性;历史信息密度可以更精确的提供个体多样性信息。在与三个先进的多目标进化算法的对比实验中,新算法的求解质量明显优于对比算法。     

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.     

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.     

均衡分布性与收敛性的协同进化多目标优化算法

为了进一步提升多目标进化算法(MOEAs)的收敛速度和解集分布性,针对变量无关问题,借助合作型协同进化模型,提出一种均衡分布性与收敛性的协同进化多目标优化算法(CMOA-BDC). CMOA-BDC 首先设置一个精英集合,采用支配关系从进化种群与精英集合中选择首层,并用拥挤距离保持其分布性;然后运用聚类将首层分类,并建立相应概率模型;最后通过模拟退火组合分布估计与遗传进化,达到协同进化.通过与经典 MOEAs 比较的结果表明, CMOA-BDC 获得的解集具有更好的收敛性和分布性.     

一种基于局部收敛估计的多目标进化算法

采用了一种基于局部收敛估计的多目标进化算法（MOEAE/LC）。在进化过程中计算连续两代归档集合群体之间的种群相似度,若在算法运行的早期其连续两代归档集的相似度小于预先设置的阈值,则认为算法有一定概率局部收敛。这时以一定概率重新初始化内部种群并且对归档集的部分个体进行变异,这样能在算法有可能陷入局部最优时产生新个体,从而提高了解集的收敛性和多样性。通过与经典的多目标算法（MOEAs）进行对比实验,实验结果表明了该算法的有效性。     

Towards faster convergence of evolutionary multi-criterion optimization algorithms using Karush Kuhn Tucker optimality based local search

Evolutionary multi-criterion optimization (EMO) algorithms emphasize non-dominated and less crowded solutions in a population iteratively until the population converges close to the Pareto optimal set. During the search process, non-dominated solutions are differentiated only by their local crowding or contribution to hypervolume or using a similar other metric. Thus, during evolution and even at the final iteration, the true convergence behavior of each non-dominated solutions from the Pareto optimal set is unknown. Recent studies have used Karush Kuhn Tucker (KKT) optimality conditions to develop a KKT Proximity Measure (KKTPM) for estimating proximity of a solution from Pareto optimal set for a multi-objective optimization problem. In this paper, we integrate KKTPM with a recently proposed EMO algorithm to enhance its convergence properties towards the true Pareto optimal front. Specifically, we use KKTPM to identify poorly converged non-dominated solutions in every generation and apply an achievement scalarizing function based local search procedure to improve their convergence. Assisted by the KKTPM, the modified algorithm is designed in a way that maintains the total number of function evaluations as low as possible while making use of local search where it is most needed. Simulations on both constrained and unconstrained multi- and many objectives optimization problems demonstrate that the hybrid algorithm significantly improves the overall convergence properties. This study brings evolutionary optimization closer to mainstream optimization field and should motivate researchers to utilize KKTPM measure further within EMO and other numerical optimization algorithms.     

求解多目标优化问题的一种多子群体进化算法

提出一种新的多目标粒子群优化（MOPSO）算法,根据多目标优化问题（MOP）的特点,将一个进化群体分成若干个子群体,利用非劣支配的概念构造全局最优区域,用以指导整个粒子群的进化．通过子群体间的信息交换．使整个群体分布更均匀,并且避免了局部最优,保证了解的多样性,通过很少的迭代次数便可得到分布均匀的Pareto有效解集．数值实验表明了该算法的有效性.     

Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm

Surrogate-assisted evolutionary optimization has proved to be effective in reducing optimization time, as surrogates, or meta-models can approximate expensive fitness functions in the optimization run. While this is a successful strategy to improve optimization efficiency, challenges arise when constructing surrogate models in higher dimensional function space, where the trade space between multiple conflicting objectives is increasingly complex. This complexity makes it difficult to ensure the accuracy of the surrogates. In this article, a new surrogate management strategy is presented to address this problem. A k-means clustering algorithm is employed to partition model data into local surrogate models. The variable fidelity optimization scheme proposed in the author's previous work is revised to incorporate this clustering algorithm for surrogate model construction. The applicability of the proposed algorithm is illustrated on six standard test problems. The presented algorithm is also examined in a three-objective stiffened panel optimization design problem to show its superiority in surrogate-assisted multi-objective optimization in higher dimensional objective function space. Performance metrics show that the proposed surrogate handling strategy clearly outperforms the single surrogate strategy as the surrogate size increases.     

Hybridizing a multi-objective simulated annealing algorithm with a multi-objective evolutionary algorithm to solve a multi-objective project scheduling problem

In this paper, a multi-objective project scheduling problem is addressed. This problem considers two conflicting, priority optimization objectives for project managers. One of these objectives is to minimize the project makespan. The other objective is to assign the most effective set of human resources to each project activity. To solve the problem, a multi-objective hybrid search and optimization algorithm is proposed. This algorithm is composed by a multi-objective simulated annealing algorithm and a multi-objective evolutionary algorithm. The multi-objective simulated annealing algorithm is integrated into the multi-objective evolutionary algorithm to improve the performance of the evolutionary-based search. To achieve this, the behavior of the multi-objective simulated annealing algorithm is self-adaptive to either an exploitation process or an exploration process depending on the state of the evolutionary-based search. The multi-objective hybrid algorithm generates a number of near non-dominated solutions so as to provide solutions with different trade-offs between the optimization objectives to project managers. The performance of the multi-objective hybrid algorithm is evaluated on nine different instance sets, and is compared with that of the only multi-objective algorithm previously proposed in the literature for solving the addressed problem. The performance comparison shows that the multi-objective hybrid algorithm significantly outperforms the previous multi-objective algorithm.