Decomposing the user-preference in multiobjective optimization

Preference information (such as the reference point) of the decision maker (DM) is often used in multiobjective optimization; however, the location of the specified reference point has a detrimental effect on the performance of multiobjective evolutionary algorithms (MOEAs). Inspired by multiobjective evolutionary algorithm-based decomposition (MOEA/D), this paper proposes an MOEA to decompose the preference information of the reference point specified by the DM into a number of scalar optimization subproblems and deals with them simultaneously (called MOEA/D-PRE). This paper presents an approach of iterative weight to map the desired region of the DM, which makes the algorithm easily obtain the desired region. Experimental results have demonstrated that the proposed algorithm outperforms two popular preference-based approaches, g-dominance and r-dominance, on continuous multiobjective optimization problems (MOPs), especially on many-objective optimization problems. Moreover, this study develops distinct models to satisfy different needs of the DM, thus providing a new way to deal with preference-based multiobjective optimization. Additionally, in terms of the shortcoming of MOEA/D-PRE, an improved MOEA/D-PRE that dynamically adjusts the size of the preferred region is proposed and has better performance on some problems.     

A Simulated Annealing-Based Multiobjective Optimization Algorithm: AMOSA

This paper describes a simulated annealing based multiobjective optimization algorithm that incorporates the concept of archive in order to provide a set of tradeoff solutions for the problem under consideration. To determine the acceptance probability of a new solution vis-a-vis the current solution, an elaborate procedure is followed that takes into account the domination status of the new solution with the current solution, as well as those in the archive. A measure of the amount of domination between two solutions is also used for this purpose. A complexity analysis of the proposed algorithm is provided. An extensive comparative study of the proposed algorithm with two other existing and well-known multiobjective evolutionary algorithms (MOEAs) demonstrate the effectiveness of the former with respect to five existing performance measures, and several test problems of varying degrees of difficulty. In particular, the proposed algorithm is found to be significantly superior for many objective test problems (e.g., 4, 5, 10, and 15 objective problems), while recent studies have indicated that the Pareto ranking-based MOEAs perform poorly for such problems. In a part of the investigation, comparison of the real-coded version of the proposed algorithm is conducted with a very recent multiobjective simulated annealing algorithm, where the performance of the former is found to be generally superior to that of the latter.     

基于ε-支配的自适应多目标进化算法

提出一种新的基于ε-支配关系的自适应多目标进化算法(AEMOEA)。在每次的进化中保留端点,并从端点集中选取一个作为父本,参加进化,弥补了ε-MOEA算法中端点易被丢掉的缺陷;在进化过程中根据存档动态地调整ε的取值,使解的分布更加均匀;当存档中个体过多时,运用ε-支配关系进行剪切,使其个体数处在合理水平。通过5个常用双目标测试函数的计算,验证了该算法在求解质量上优于ε-MOEA、NAGA-II以及SPEA-2等主流多目标算法。     

基于改进K 支配排序的高维多目标进化算法

为提高4目标以上高维多目标优化问题的求解性能,提出一种基于改进K支配排序的高维多目标进化算法(KS-MODE).该算法针对K支配的支配关系和排序方法进行改进,避免循环支配并增强选择压力;设计新的全局密度估计方法提高局部密度估计精确性;设计新的精英选择策略和适应度值评价函数;采用CAO局部搜索算子加速收敛.在4～30个目标标准测试函数上的实验结果表明,KS-MODE能够在保证解集分布性的同时大幅提升收敛性和稳定性,能够有效求解高维多目标优化问题.     

Improving the efficiency of ?-dominance based grids

In this paper, we deal with the problem of handling solutions in an external archive with the use of a relaxed form of Pareto dominance called ?-dominance and a variation of it called pa?-dominance. These two relaxed forms of Pareto dominance have been used as archiving strategies in some multi-objective evolutionary algorithms (MOEAs). The main objective of this work is to improve the ?-dominance based schemes to handle nondominated solutions, or to retain nondominated solutions in an external archive. Thus, our main contribution is to add an extra objective function only at the time of accepting a nondominated solution into the external archive, in order to preserve some solutions which are normally lost when using any of the aforementioned relaxed forms of Pareto dominance. Such a proposal is inexpensive (computationally speaking) and quite effective, since it is able to produce Pareto fronts of much better quality than the aforementioned archiving techniques.     

Simultaneous topology and sizing optimization of a water distribution network using a hybrid multiobjective evolutionary algorithm

This paper proposes a new direction for design optimization of a water distribution network (WDN). The new approach introduces an optimization process to the conceptual design stage of a WDN. The use of multiobjective evolutionary algorithms (MOEAs) for simultaneous topology and sizing design of piping networks is presented. The design problem includes both topological and sizing design variables while the objective functions are network cost and total head loss in pipes. The numerical technique, called a network repairing technique (NRT), is proposed to overcome difficulties in operating MOEAs for network topological design. The problem is then solved by using a number of established and newly developed MOEAs. Also, two new MOEAs namely multiobjective real code population-based incremental learning (RPBIL) and a hybrid algorithm of RPBIL with differential evolution (termed RPBIL–DE) are proposed to tackle the design problems. The optimum results obtained are illustrated and compared. It is shown that the proposed network repairing technique is an efficient and effective tool for topological design of WDNs. Based on the hypervolume indicator, the proposed RPBIL–DE is among the best MOEA performers.     

Combining convergence and diversity in evolutionary multiobjective optimization

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Pareto-optimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept of epsilon-dominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modifications to the baseline algorithm are also suggested. The concept of epsilon-dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.     

Using unconstrained elite archives for multiobjective optimization

Multiobjective evolutionary algorithms (MOEAs) have been the subject of numerous studies over the past 20 years. Recent work has highlighted the use of an active archive of elite, nondominated solutions to improve the optimization speed of these algorithms. However, preserving all elite individuals is costly in time (due to the linear comparison with all archived solutions needed before a new solution can be inserted into the archive). Maintaining an elite population of a fixed maximum size (by clustering or other means) alleviates this problem, but can cause retreating (or oscillitory) and shrinking estimated Pareto fronts - which can affect the efficiency of the search process. New data structures are introduced to facilitate the use of an unconstrained elite archive, without the need for a linear comparison to the elite set for every new individual inserted. The general applicability of these data structures is shown by their use in an evolution-strategy-based MOEA and a genetic-algorithm-based MOEA. It is demonstrated that MOEAs using the new data structures run significantly faster than standard, unconstrained archive MOEAs, and result in estimated Pareto fronts significantly ahead of MOEAs using a constrained archive. It is also shown that the use of an unconstrained elite archive permits robust criteria for algorithm termination to be used, and that the use of the data structure can also be used to increase the speed of algorithms using /spl epsi/-dominance methods.     

The balance between proximity and diversity in multiobjective evolutionary algorithms

Over the last decade, a variety of evolutionary algorithms (EAs) have been proposed for solving multiobjective optimization problems. Especially more recent multiobjective evolutionary algorithms (MOEAs) have been shown to be efficient and superior to earlier approaches. An important question however is whether we can expect such improvements to converge onto a specific efficient MOEA that behaves best on a large variety of problems. In this paper, we argue that the development of new MOEAs cannot converge onto a single new most efficient MOEA because the performance of MOEAs shows characteristics of multiobjective problems. While we point out the most important aspects for designing competent MOEAs in this paper, we also indicate the inherent multiobjective tradeoff in multiobjective optimization between proximity and diversity preservation. We discuss the impact of this tradeoff on the concepts and design of exploration and exploitation operators. We also present a general framework for competent MOEAs and show how current state-of-the-art MOEAs can be obtained by making choices within this framework. Furthermore, we show an example of how we can separate nondomination selection pressure from diversity preservation selection pressure and discuss the impact of changing the ratio between these components.     

Solving multiobjective problems using cat swarm optimization

This paper proposes a new multiobjective evolutionary algorithm (MOEA) by extending the existing cat swarm optimization (CSO). It finds the nondominated solutions along the search process using the concept of Pareto dominance and uses an external archive for storing them. The performance of our proposed approach is demonstrated using standard test functions. A quantitative assessment of the proposed approach and the sensitivity test of different parameters is carried out using several performance metrics. The simulation results reveal that the proposed approach can be a better candidate for solving multiobjective problems (MOPs).     

Multiobjective evolutionary algorithm based on multimethod with dynamic resources allocation

In the last two decades, multiobjective optimization has become main stream and various multiobjective evolutionary algorithms (MOEAs) have been suggested in the field of evolutionary computing (EC) for solving hard combinatorial and continuous multiobjective optimization problems. Most MOEAs employ single evolutionary operators such as crossover, mutation and selection for population evolution. In this paper, we suggest a multiobjective evolutionary algorithm based on multimethods (MMTD) with dynamic resource allocation for coping with continuous multi-objective optimization problems (MOPs). The suggested algorithm employs two well known population based stochastic algorithms namely MOEA/D and NSGA-II as constituent algorithms for population evolution with a dynamic resource allocation scheme. We have examined the performance of the proposed MMTD on two different MOPs test suites: the widely used ZDT problems and the recently formulated test instances for the special session on MOEAs competition of the 2009 IEEE congress on evolutionary computation (CEC’09). Experimental results obtained by the suggested MMTD are more promising than those of some state-of-the-art MOEAs in terms of the inverted generational distance (IGD)-metric on most test problems.     

Comparison of multi-objective evolutionary algorithms in hybrid Kansei engineering system for product form design

Understanding the affective needs of customers is crucial to the success of product design. Hybrid Kansei engineering system (HKES) is an expert system capable of generating products in accordance with the affective responses. HKES consists of two subsystems: forward Kansei engineering system (FKES) and backward Kansei engineering system (BKES). In previous studies, HKES was based primarily on single-objective optimization, such that only one optimal design was obtained in a given simulation run. The use of multi-objective evolutionary algorithm (MOEA) in HKES was only attempted using the non-dominated sorting genetic algorithm-II (NSGA-II), such that very little work has been conducted to compare different MOEAs. In this paper, we propose an approach to HKES combining the methodologies of support vector regression (SVR) and MOEAs. In BKES, we constructed predictive models using SVR. In FKES, optimal design alternatives were generated using MOEAs. Representative designs were obtained using fuzzy c-means algorithm for clustering the Pareto front into groups. To enable comparison, we employed three typical MOEAs: NSGA-II, the Pareto envelope-based selection algorithm-II (PESA-II), and the strength Pareto evolutionary algorithm-2 (SPEA2). A case study of vase form design was provided to demonstrate the proposed approach. Our results suggest that NSGA-II has good convergence performance and hybrid performance; in contrast, SPEA2 provides the strong diversity required by designers. The proposed HKES is applicable to a wide variety of product design problems, while providing creative design ideas through the exploration of numerous Pareto optimal solutions.     

Tuning of PID controller based on a multiobjective genetic algorithm applied to a robotic manipulator

Most controllers optimization and design problems are multiobjective in nature, since they normally have several (possibly conflicting) objectives that must be satisfied at the same time. Instead of aiming at finding a single solution, the multiobjective optimization methods try to produce a set of good trade-off solutions from which the decision maker may select one. Several methods have been devised for solving multiobjective optimization problems in control systems field. Traditionally, classical optimization algorithms based on nonlinear programming or optimal control theories are applied to obtain the solution of such problems. The presence of multiple objectives in a problem usually gives rise to a set of optimal solutions, largely known as Pareto-optimal solutions. Recently, Multiobjective Evolutionary Algorithms (MOEAs) have been applied to control systems problems. Compared with mathematical programming, MOEAs are very suitable to solve multiobjective optimization problems, because they deal simultaneously with a set of solutions and find a number of Pareto optimal solutions in a single run of algorithm. Starting from a set of initial solutions, MOEAs use iteratively improving optimization techniques to find the optimal solutions. In every iterative progress, MOEAs favor population-based Pareto dominance as a measure of fitness. In the MOEAs context, the Non-dominated Sorting Genetic Algorithm (NSGA-II) has been successfully applied to solving many multiobjective problems. This paper presents the design and the tuning of two PID (Proportional–Integral–Derivative) controllers through the NSGA-II approach. Simulation numerical results of multivariable PID control and convergence of the NSGA-II is presented and discussed with application in a robotic manipulator of two-degree-of-freedom. The proposed optimization method based on NSGA-II offers an effective way to implement simple but robust solutions providing a good reference tracking performance in closed loop.     

一种基于混合高斯模型的多目标进化算法

目前,大多数多目标进化算法采用为单目标优化所设计的重组算子.通过证明或实验分析了几个典型的单目标优化重组算子并不适合某些多目标优化问题.提出了基于分解技术和混合高斯模型的多目标优化算法（multiobjective evolutionary algorithm based on decomposition and mixture Gaussian models,简称MOEA/D-MG）.该算法首先采用一个改进的混合高斯模型对群体建模并采样产生新个体,然后利用一个贪婪策略来更新群体.针对具有复杂Pareto前沿的多目标优化问题的测试结果表明,对给定的大多数测试题,该算法具有良好的效果.     

Effective Evolutionary Algorithms for Many-Specifications Attainment: Application to Air Traffic Control Tracking Filters

This paper addresses a real-world engineering design requiring the application of effective and global optimization techniques. The problem it deals with is the design of nonlinear tracking filters under up to several hundreds of performance specifications. The suitability of different evolutionary computation techniques for solving multiobjective problems is explored, contrasting the performance achieved with recent multiobjective evolutionary algorithm (MOEAs) proposals and different aggregation schemes. In particular, a new scheme is proposed to build a fitness function based on an operator that selects worst cases of multiple specifications in different situations. They have been evaluated in the design of an air traffic control (ATC) tracking filter that should accomplish a specific normative with 264 specifications. Results show their performance in terms of effectiveness and computational load, comparing their capability to scale the problem with respect to problem size.     

基于新的网格存优策略的多目标归档算法

网格方法被多个进化算法用来保持解集的分布性。基于ε支配概念的ε-MOEA本质上也是基于网格策略的。虽然ε-MOEA通常情况下都能在算法性能的各方面之间取得较为合理的折衷,但是由于其存在固有缺陷,很多时候表现出不容忽视的问题——当PF_true对某一维的变化率在该维不同区域的差异较大时,解集中边界个体或代表性个体丢失——严重影响解集的分布性。针对这一问题,定义了一种新的δ支配概念和虚拟“最优点”的概念,提出了一种新的网格存优策略,并将之应用于更新进化多目标归档算法的归档集。实验结果显示,基于新的存优策略的进化多目标归档算法（δ-MOEA）具有良好的性能,尤其在分布性方面比NSGA2和ε-MOEA好得多。     

Using genetic algorithm and TOPSIS technique for multiobjective transportation problem: a hybrid approach

A hybrid approach to solve the multiobjective transportation problem (TP) is presented. The TP as a special type of the network optimization problems that has the special data structure in solution characterized as transportation graph. In encoding TP, we introduce a new chromosome's structure which is adopted as it is capable of representing all possible feasible solutions. Also, in order to keep the feasibility of the chromosome, the crossover and the mutation were modified. The proposed approach maintains a finite-sized archive of non-dominated solutions which gets iteratively updated in the presence of new solutions based on the concept of ?-dominance. Moreover, to help the decision maker to extract the best compromise solution from a finite set of alternatives, a technique for order performance by similarity to ideal solution (TOPSIS) method is adopted. Numerical simulations show the effectiveness and efficiency of the proposed approach.     

Interactive Evolutionary Multiobjective Search and Optimization of Set-Based Concepts

This paper deals with interactive concept-based multiobjective problems (IC-MOPs) and their solution by an evolutionary computation approach. The presented methodology is motivated by the need to support engineers during the conceptual design stage. IC-MOPs are based on a nontraditional concept-based approach to search and optimization. It involves conceptual solutions, which are represented by sets of particular solutions, with each concept having a one-to-many relation with the objective space. Such a set-based concept representation is most suitable for human–computer interaction. Here, a fundamental type of IC-MOPs, namely, the Pareto-directed one, is formally defined, and its solution is presented. Next, a new interactive concept-based multiobjective evolutionary algorithm is introduced, and measures to assess its resulting fronts are devised. Finally, the proposed approach and the suggested search algorithm are studied using both academic test functions and an engineering problem.      

基于Pareto熵的多目标粒子群优化算法

粒子群优化算法因形式简洁、收敛快速和参数调节机制灵活等优点,同时一次运行可得到多个解,且能逼近非凸或不连续的Pareto最优前端,因而被认为是求解多目标优化问题最具潜力的方法之一.但当粒子群优化算法从单目标问题扩展到多目标问题时,Pareto最优解集的存储与维护、全局和个体最优解的选择以及开发与开采的平衡等问题亦随之出现.通过目标空间变换方法,采用Pareto前端在被称为平行格坐标系统的新目标空间中的分布熵及差熵评估种群的多样性及进化状态,并以此为反馈信息来设计进化策略,使得算法能够兼顾近似Pareto前端的收敛性和多样性.同时,引入格占优和格距离密度的概念来评估Pareto最优解的个体环境适应度,以此建立外部档案更新方法和全局最优解选择机制,最终形成了基于Pareto熵的多目标粒子群优化算法.实验结果表明：在IGD性能指标上,与另外8种对等算法相比,该算法在由ZDT和DTLZ系列组成的12个多目标测试问题集中表现出了显著的性能优势.     

基于ε占优的自适应多目标粒子群算法

针对粒了群算法求解多目标问题极易收敛到伪Pareto前沿(等价于单目标优化问题中的局部最优解),并且收敛速度较慢的问题,提出一种ε占优的自适应多目标粒子群算法(εDMOPSO)..在εDMOPSO算法中,每个粒子的邻居根据粒了的运行动态地组建,且粒了的速度小由其邻居中运行最好的粒予来调整,而是由其所有邻居共同调整.同时...