基于在线感知Pareto前沿划分目标空间的多目标进化优化

多目标进化优化是求解多目标优化问题的可行方法.但是, 由于没有准确感知并充分利用问题的Pareto前沿, 已有方法难以高效求解复杂的多目标优化问题.本文提出一种基于在线感知Pareto前沿划分目标空间的多目标进化优化方法, 以利用感知的结果, 采用有针对性的进化优化方法求解多目标优化问题.首先, 根据个体之间的拥挤距离与给定阈值的关系感知优化问题的Pareto前沿上的间断点, 并基于此将目标空间划分为若干子空间; 然后, 在每一子空间中采用MOEA/D (Multi-objective evolutionary algorithm based on decomposition)得到一个外部保存集; 最后, 基于所有外部保存集生成问题的Pareto解集.将提出的方法应用于15个基准数值函数优化问题, 并与NSGA-Ⅱ、RPEA、MOEA/D、MOEA/DPBI、MOEA/D-STM和MOEA/D-ACD等比较.结果表明, 提出的方法能够产生收敛和分布性更优的Pareto解集, 是一种非常有竞争力的方法.     

基于幂变换的多目标进化算法MOEA?D权重设计方法

在多目标最优化问题中,如何求解一组均匀散布在前沿界面上的有效解具有重要意义.MOEA?D是最近出现的一种杰出的多目标进化算法,当前沿界面的形状是某种已知的类型时,MOEA?D使用高级分解的方法容易求出均匀散布在前沿界面上的有效解.然而,多目标优化问题的前沿界面的形状通常是未知的.为了使MOEA?D能求出一般多目标优化问题的均匀散布的有效解,利用幂函数对目标进行数学变换,使变换后的多目标优化问题的前沿界面在算法的进化过程中逐渐接近希望得到的形状,提出了一种求解一般的多目标优化问题的MOEA?D算法的权重设计方法,并且讨论了经过数学变换后前沿界面的保距性问题.采用建议的权重设计方法,MOEA?D更容易求出一般的多目标优化问题均匀散布的有效解.数值结果验证了算法的有效性.     

Utopian point based decomposition for multi-objective optimization problems with complicated Pareto fronts

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) has been considered as a promising method for solving multi-objective optimization problems (MOPs). It devotes most of its effort on convergence by optimizing a set of scalar optimization subproblems in a collaborative manner, while maintaining the diversity by using a set of uniformly distributed weight vectors. However, more recent studies illustrated that MOEA/D faces difficulties on MOPs with complicated Pareto fronts, mainly because the uniformity of weight vectors no longer lead to an evenly scattered approximation of the Pareto fronts in these cases. To remedy this, we suggest replacing the ideal point in the reciprocal Tchebycheff decomposition method with a more optimistic utopian point, with the aim of alleviating the sensitivity of MOEA/D to the Pareto front shape of MOPs. Experimental studies on benchmark and real-world problems have shown that such simple modification can significantly improve the performances of MOEA/D with reciprocal Tchebycheff decomposition on MOPs with complicated Pareto fronts.     

BSTBGA: A hybrid genetic algorithm for constrained multi-objective optimization problems

Most of the existing multi-objective genetic algorithms were developed for unconstrained problems, even though most real-world problems are constrained. Based on the boundary simulation method and trie-tree data structure, this paper proposes a hybrid genetic algorithm to solve constrained multi-objective optimization problems (CMOPs). To validate our approach, a series of constrained multi-objective optimization problems are examined, and we compare the test results with those of the well-known NSGA-II algorithm, which is representative of the state of the art in this area. The numerical experiments indicate that the proposed method can clearly simulate the Pareto front for the problems under consideration.     

Multiobjective Evolutionary Algorithm With Controllable Focus on the Knees of the Pareto Front

The optimal solutions of a multiobjective optimization problem correspond to a nondominated front that is characterized by a tradeoff between objectives. A knee region in this Pareto-optimal front, which is visually a convex bulge in the front, is important to decision makers in practical contexts, as it often constitutes the optimum in tradeoff, i.e., substitution of a given Pareto-optimal solution with another solution on the knee region yields the largest improvement per unit degradation. This paper presents a selection scheme that enables a multiobjective evolutionary algorithm (MOEA) to obtain a nondominated set with controllable concentration around existing knee regions of the Pareto front. The preference-based focus is achieved by optimizing a set of linear weighted sums of the original objectives, and control of the extent of the focus is attained by careful selection of the weight set based on a user-specified parameter. The fitness scheme could be easily adopted in any Pareto-based MOEA with little additional computational cost. Simulations on various two- and three-objective test problems demonstrate the ability of the proposed method to guide the population toward existing knee regions on the Pareto front. Comparison with general-purpose Pareto based MOEA demonstrates that convergence on the Pareto front is not compromised by imposing the preference-based bias. The performance of the method in terms of an additional performance metric introduced to measure the accuracy of resulting convergence on the desired regions validates the efficacy of the method.      

Multiobjective evolutionary algorithms for electric power dispatch problem

The potential and effectiveness of the newly developed Pareto-based multiobjective evolutionary algorithms (MOEA) for solving a real-world power system multiobjective nonlinear optimization problem are comprehensively discussed and evaluated in this paper. Specifically, nondominated sorting genetic algorithm, niched Pareto genetic algorithm, and strength Pareto evolutionary algorithm (SPEA) have been developed and successfully applied to an environmental/economic electric power dispatch problem. A new procedure for quality measure is proposed in this paper in order to evaluate different techniques. A feasibility check procedure has been developed and superimposed on MOEA to restrict the search to the feasible region of the problem space. A hierarchical clustering algorithm is also imposed to provide the power system operator with a representative and manageable Pareto-optimal set. Moreover, an approach based on fuzzy set theory is developed to extract one of the Pareto-optimal solutions as the best compromise one. These multiobjective evolutionary algorithms have been individually examined and applied to the standard IEEE 30-bus six-generator test system. Several optimization runs have been carried out on different cases of problem complexity. The results of MOEA have been compared to those reported in the literature. The results confirm the potential and effectiveness of MOEA compared to the traditional multiobjective optimization techniques. In addition, the results demonstrate the superiority of the SPEA as a promising multiobjective evolutionary algorithm to solve different power system multiobjective optimization problems.     

SP-MEC算法的收敛性分析

进化算法求解多目标优化问题具有独特的优势。SP-MEC是一种新的利用思维进化算法(MEC)解决多目标优化问题的算法,数值实验结果验证了它的可行性与有效性。文章利用概率论的基本理论对其收敛性进行分析,提出局部Pareto最优解集、局部Pareto最优态集及趋同过程产生的序列强收敛的概念,证明了在满足一定条件下趋同过程产生的序列强收敛于局部Pareto最优态集。     

Pareto-MEC算法及其收敛性分析

介绍了一种新的多目标进化算法——Pareto-MEC。将基本MEC和Pareto思想结合起来处理多目标问题。提出了局部Pareto最优解集与局部Pareto最优态集概念，并利用概率论的基本理论证明了趋同过程产生的序列强收敛于局部Pareto最优态集。数值试验验证了Pareto-MEC算法的有效性。     

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.     

面向多目标优化的适应度共享免疫克隆算法

多目标优化的日标在于使得解集能够快速的逼近真实Pareto前沿.针对解的分布性问题,以免疫克隆算法为框架,引入适应度共享策略,提出了一种新的具有良好分布性保持的多目标优化进化算法;算法建立外部群体以保存非支配解,以Pareto优和共亨适应度作为外部群体更新与激活抗体选择的双重标准.为了增强算法对决策空间的开发能力,引入...     

Incorporation of implicit decision-maker preferences in multi-objective evolutionary optimization using a multi-criteria classification method

Nowadays, most Multi-Objective Evolutionary Algorithms (MOEA) concentrate mainly on searching for an approximation of the Pareto frontier to solve a multi-objective optimization problem. However, finding this set does not completely solve the problem. The decision-maker (DM) still has to choose the best compromise solution from that set. But as the number of criteria increases, several important difficulties arise in performing this task. Identifying the Region of Interest (ROI), according to the DM’s preferences, is a promising alternative that would facilitate the selection process. This paper approaches the incorporation of preferences into a MOEA in order to characterize the ROI by a multi-criteria classification method. This approach is called Hybrid Multi-Criteria Sorting Genetic Algorithm and is composed of two phases. First, a metaheuristic is used to generate a small set of solutions that are classified in ordered categories by the DM. Thus, the DM’s preferences will be reflected indirectly in this set. In the second phase, a multi-criteria sorting method is combined with an evolutionary algorithm. The first one is used to classify new solutions. Those classified as ‘satisfactory’ are used for creating a selective pressure towards the ROI. The effectiveness of our method was proved in nine instances of a public project portfolio problem. The obtained results indicate that our approach achieves a good characterization of the ROI, and outperforms the standard NSGA-II in simple and complex problems. Also, these results confirm that our approach is able to deal with many-objective problems.     

混合分解和强度帕累托多目标进化算法

在多目标进化优化中,使用分解策略的基于分解的多目标进化算法(MOEA/D)时间复杂度低,使用〖BP(〗强度帕累托策略的〖BP)〗强度帕累托进化算法-2(SPEA2)能得到分布均匀的解集。结合这两种策略,提出一种新的多目标进化算法用于求解具有复杂、不连续的帕累托前沿的多目标优化问题(MOP)。首先,利用分解策略快速逼近帕累托前沿;然后,利用强度帕累托策略使解集均匀分布在帕累托前沿,利用解集重置分解策略中的权重向量集,使其适配于特定的帕累托前沿;最后,利用分解策略进一步逼近帕累托前沿。使用的反向世代距离(IGD)作为度量标准,将新算法与MOEA/D、SPEA2和paλ-MOEA/D在12个基准问题上进行性能对比。实验结果表明该算法性能在7个基准问题上最优,在5个基准问题上接近于最优,且无论MOP的帕累托前沿是简单或复杂、连续或不连续的,该算法均能生成分布均匀的解集。     

Evolutionary algorithms in multiply-specified engineering. The MOEAs and WCES strategies

This paper addresses multi-objective optimization from the viewpoint of real-world engineering designs with lots of specifications, where robust and global optimization techniques need to be applied. The problem used to illustrate the process is the design of non-linear control systems with hundreds of performance specifications. The performance achieved with a recent multi-objective evolutionary algorithm (MOEA) is compared with a proposed scheme to build a robust fitness function aggregation. The proposed strategy considers performances in the worst situations: worst-case combination evolution strategy (WCES), and it is shown to be robust against the dimensionality of specifications. A representative MOEA, SPEA-2, achieved a satisfactory performance with a moderate number of specifications, but required an exponential increase in population size as more specifications were added. This becomes impractical beyond several hundreds. WCES scales well against the problem size, since it exploits the similar behaviour of magnitudes evaluated under different situations and searches similar trade-offs for correlated objectives. Both approaches have been thoroughly characterized considering increasing levels of complexity, different design problems, and algorithm configurations.     

MOEA/D线性插入方向向量策略研究

MOEA/D具有良好的收敛性、均匀的分布性、求解效率高等优点,普遍应用于求解多目标优化问题.然而对于Pareto前端复杂的多目标优化问题,预先设定均匀的权重向量并不能够维持Pareto最优解集的良好分布性.本文,首先分析均匀分布的权重向量、均匀分布的搜索方向二者与均匀分布的解集之间的关系,提出一种新的权重向量设置方式;其次基于进化过程中解集的分布,提出线性插入搜索方向策略,并将其转换为对应的权重向量,同时在MOEA/D中周期性应用该策略调整搜索方向,获取分布均匀的解集;最后将该算法在WFG系列测试问题上进行性能测试,并采用世代距离指标(GD)、Spacing指标(S)、超体积指标(HV)对算法收敛性和多样性进行对比分析,实验结果表明,与原始的MOEA/D、使用均匀分布的搜索方向MOEA/D、使用预处理的M OEA/D、M OEA/D-DU相比,改进的算法求出解集的多样性极大提高,收敛性明显增强,解集的整体质量显著提高.     

记忆增强的动态多目标分解进化算法

现实世界中的一些多目标优化问题经常受动态环境影响而不断发生变化,要求优化算法不断地及时跟踪时变的Pareto 最优解集.提出了一种记忆增强的动态多目标分解进化算法.将动态多目标优化问题分解为若干个动态单目标优化子问题并同时优化这些子问题,以便快速逼近Pareto 最优解集.给出了一个改进的环境变化检测算子,以便更好地检测环境变化.设计了一种基于子问题的串式记忆方法,利用过去类似环境下搜索到的最优解来有效地响应新的环境变化.在8 个标准的测试问题上,将新算法与其他3 种记忆增强的动态进化多目标优化算法进行了实验比较.结果表明,新算法比其他3 种算法具有更快的运行速度、更强的记忆能力与鲁棒性能,并且新算法所获得的解集还具有更好的收敛性与分布性.     

A decomposition based memetic algorithm for multi-objective vehicle routing problem with time windows

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) provides an excellent algorithmic framework for solving multi-objective optimization problems. It decomposes a target problem into a set of scalar sub-problems and optimizes them simultaneously. Due to its simplicity and outstanding performance, MOEA/D has been widely studied and applied. However, for solving the multi-objective vehicle routing problem with time windows (MO-VRPTW), MOEA/D faces a difficulty that many sub-problems have duplicated best solutions. It is well-known that MO-VRPTW is a challenging problem and has very few Pareto optimal solutions. To address this problem, a novel selection operator is designed in this work to enhance the original MOEA/D for dealing with MO-VRPTW. Moreover, three local search methods are introduced into the enhanced algorithm. Experimental results indicate that the proposed algorithm can obtain highly competitive results on Solomon׳s benchmark problems. Especially for instances with long time windows, the proposed algorithm can obtain more diverse set of non-dominated solutions than the other algorithms. The effectiveness of the proposed selection operator is also demonstrated by further analysis.     

基于分解和差分进化的多目标粒子群优化算法

为了提高多目标优化算法解集的分布性和收敛性,提出一种基于分解和差分进化的多目标粒子群优化算法(dMOPSO-DE).该算法通过提出方向角产生一组均匀的方向向量,确保粒子分布的均匀性;引入隐式精英保持策略和差分进化修正机制选择全局最优粒子,避免种群陷入局部最优Pareto前沿;采用粒子重置策略保证群体的多样性.与非支配排序(NSGA-II)算法、多目标粒子群优化(MOPSO)算法、分解多目标粒子群优化(dMOPSO)算法和分解多目标进化-差分进化(MOEA/D-DE)算法进行比较,实验结果表明,所提出算法在求解多目标优化问题时具有良好的收敛性和多样性.     

Multi-objective optimization of double suction centrifugal pump using Kriging metamodels

This paper proposes a new multi-objective optimization method for a family of double suction centrifugal pumps with various blade shapes, using a Simulation-Kriging model-Experiment (SKE) approach. The Kriging metamodel is established to approximate the characteristic performance functions of a pump, namely, the efficiency and required net positive suction head (NPSHr). Hence, the two objectives are to maximize the efficiency and simultaneously to minimize NPSHr. The Non-dominated Sorting Genetic Algorithm II (NSGA II) and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) have been applied to the multi-objective optimization problem, respectively. The Pareto solution set is obtained by a more effective and efficient manner of the two multi-objective optimization algorithms. A tradeoff optimal design point is selected from the Pareto solution set by means of a robust design based on Monte Carlo simulations, and the optimal solution is further compared with the value of the physical prototype test. The results show that the solution of the proposed multi-objective optimization method is in line with the experiment test.     

Multi-objective node deployment in WSNs: In search of an optimal trade-off among coverage,lifetime, energy consumption,and connectivity

The increased demand of Wireless Sensor Networks (WSNs) in different areas of application have intensified studies dedicated to the deployment of sensor nodes in recent past. For deployment of sensor nodes some of the key objectives that need to be satisfied are coverage of the area to be monitored, net energy consumed by the WSN, lifetime of the network, and connectivity and number of deployed sensors. In this article the sensor node deployment task has been formulated as a constrained multi-objective optimization (MO) problem where the aim is to find a deployed sensor node arrangement to maximize the area of coverage, minimize the net energy consumption, maximize the network lifetime, and minimize the number of deployed sensor nodes while maintaining connectivity between each sensor node and the sink node for proper data transmission. We assume a tree structure between the deployed nodes and the sink node for data transmission. Our method employs a recently developed and very competitive multi-objective evolutionary algorithm (MOEA) known as MOEA/D-DE that uses a decomposition approach for converting the problem of approximation of the Pareto fronts (PF) into a number of single-objective optimization problems. This algorithm employs differential evolution (DE), one of the most powerful real parameter optimizers in current use, as its search method. The original MOEA/D has been modified by introducing a new fuzzy dominance based decomposition technique. The algorithm introduces a fuzzy Pareto dominance concept to compare two solutions and uses the scalar decomposition method only when one of the solutions fails to dominate the other in terms of a fuzzy dominance level. We have compared the performance of the resulting algorithm, called MOEA/DFD, with the original MOEA/D-DE and another very popular MOEA called Non-dominated Sorting Genetic Algorithm (NSGA-II). The best trade-off solutions from MOEA/DFD based node deployment scheme have also been compared with a few single-objective node deployment schemes based on the original DE, an adaptive DE-variant (JADE), original particle swarm optimization (PSO), and a state-of-the art variant of PSO (Comprehensive Learning PSO). In all the test instances, MOEA/DFD performs better than all other algorithms. Also the proposed multi-objective formulation of the problem adds more flexibility to the decision maker for choosing the necessary threshold of the objectives to be satisfied.     

A hybrid genetic algorithm for constrained multi-objective optimization under uncertainty and target matching problems

This work presents a new approach for interval-based uncertainty analysis. The proposed approach integrates a local search strategy as the worst-case-scenario technique of anti-optimization with a constrained multi-objective genetic algorithm. Anti-optimization is a term for an approach to safety factors in engineering structures which is described as pessimistic and searching for least favorable responses, in combination with optimization techniques but in contrast to probabilistic approaches. The algorithm is applied and evaluated to be efficient and effective in producing good results via target matching problems: a simulated topology and shape optimization problem where a ‘target’ geometry set is predefined as the Pareto optimal solution and a constrained multiobjective optimization problem formulated such that the design solutions will evolve and converge towards the target geometry set.