Multi-objective genetic algorithms: problem difficulties and construction of test problems.

In this paper, we study the problem features that may cause a multi-objective genetic algorithm (GA) difficulty in converging to the true Pareto-optimal front. Identification of such features helps us develop difficult test problems for multi-objective optimization. Multi-objective test problems are constructed from single-objective optimization problems, thereby allowing known difficult features of single-objective problems (such as multi-modality, isolation, or deception) to be directly transferred to the corresponding multi-objective problem. In addition, test problems having features specific to multi-objective optimization are also constructed. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multi-objective optimization.     

Efficiency determination of induction motors using multi-objective evolutionary algorithms

This paper introduces a method based on multi-objective evolutionary algorithms for the determination of in-service induction motor efficiency. In general, the efficiency is determined by accumulating multiple objectives into one objective by a linear combination and optimizing the resulting single-objective problem. The approach has some drawbacks such that exact information about solution alternatives will not be readily visible. In this paper the multi-objective evolutionary optimization algorithms, the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) and Strength Pareto Evolutionary Algorithm-2 (SPEA2), are successfully applied to the efficiency determination problem in induction motor. The performances of algorithms are compared on the basis of the obtained results.     

一种基于多样性度量的多目标化进化算法

针对如何通过附加的方法对多目标化问题进行理论分析,提出并证明了选择附加函数的3个前提条件.提出一种多目标化进化算法,根据种群中个体的多样性度量进行多目标化,并采用改进的非劣分类遗传算法对构造所得的多目标优化问题进行多目标优化.在静态和动态两种环境下进行算法性能验证,结果表明,在种群多样性保持、处理欺骗问题、动态环境下的适应能力等方面,所提算法明显优于其他同类算法.     

Plateaus can be harder in multi-objective optimization

In recent years a lot of progress has been made in understanding the behavior of evolutionary computation methods for single- and multi-objective problems. Our aim is to analyze the diversity mechanisms that are implicitly used in evolutionary algorithms for multi-objective problems by rigorous runtime analyses. We show that, even if the population size is small, the runtime can be exponential where corresponding single-objective problems are optimized within polynomial time. To illustrate this behavior we analyze a simple plateau function in a first step and extend our result to a class of instances of the well-known SetCover problem.     

基于多目标进化算法的MOEA/D权重向量产生方法

在进化多目标优化研究领域,多目标优化是指对含有2个及以上目标的多目标问题的同时优化,其在近些年来受到越来越多的关注。随着MOEA/D的提出,基于聚合的多目标进化算法得到越来越多的研究,对MOEA/D算法的改进已有较多成果,但是很少有成果研究MOEA/D中权重的产生方法。提出一种使用多目标进化算法产生任意多个均匀分布的权重向量的方法,将其应用到MOEA/D,MSOPS和NSGA-III中,对这3个经典的基于聚合的多目标进化算法进行系统的比较研究。通过该类算法在DTLZ测试集、多目标旅行商问题MOTSP上的优化结果来分别研究该类算法在连续性问题、组合优化问题上的优化能力,以及使用矩形测试问题使得多目标进化算法的优化结果在决策空间可视化。实验结果表明,没有一个算法能适用于所有特性的问题。然而,MOEA/D采用不同聚合函数的两个算法MOEA/D_Tchebycheff和MOEA/D_PBI在多数情况下的性能比MSOPS和NSGA-III更好。     

DOPGA: a new fitness assignment scheme for multi-objective evolutionary algorithms

In this article, a new fitness assignment scheme to evaluate the Pareto-optimal solutions for multi-objective evolutionary algorithms is proposed. The proposed DOmination Power of an individual Genetic Algorithm (DOPGA) method can order the individuals in a form in which each individual (the so-called solution) could have a unique rank. With this new method, a multi-objective problem can be treated as if it were a single-objective problem without drastically deviating from the Pareto definition. In DOPGA, relative position of a solution is embedded into the fitness assignment procedures. We compare the performance of the algorithm with two benchmark evolutionary algorithms (Strength Pareto Evolutionary Algorithm (SPEA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2)) on 12 unconstrained bi-objective and one tri-objective test problems. DOPGA significantly outperforms SPEA on all test problems. DOPGA performs better than SPEA2 in terms of convergence metric on all test problems. Also, Pareto-optimal solutions found by DOPGA spread better than SPEA2 on eight of 13 test problems.     

Integrating Conjugate Gradients Into Evolutionary Algorithms for Large-Scale Continuous Multi-Objective Optimization

Large-scale multi-objective optimization problems (LSMOPs) pose challenges to existing optimizers since a set of well-converged and diverse solutions should be found in huge search spaces. While evolutionary algorithms are good at solving small-scale multi-objective optimization problems, they are criticized for low efficiency in converging to the optimums of LSMOPs. By contrast, mathematical programming methods offer fast convergence speed on large-scale single-objective optimization problems, but they have difficulties in finding diverse solutions for LSMOPs. Currently, how to integrate evolutionary algorithms with mathematical programming methods to solve LSMOPs remains unexplored. In this paper, a hybrid algorithm is tailored for LSMOPs by coupling differential evolution and a conjugate gradient method. On the one hand, conjugate gradients and differential evolution are used to update different decision variables of a set of solutions, where the former drives the solutions to quickly converge towards the Pareto front and the latter promotes the diversity of the solutions to cover the whole Pareto front. On the other hand, objective decomposition strategy of evolutionary multi-objective optimization is used to differentiate the conjugate gradients of solutions, and the line search strategy of mathematical programming is used to ensure the higher quality of each offspring than its parent. In comparison with state-of-the-art evolutionary algorithms, mathematical programming methods, and hybrid algorithms, the proposed algorithm exhibits better convergence and diversity performance on a variety of benchmark and real-world LSMOPs.      

Introducing robustness in multi-objective optimization

In optimization studies including multi-objective optimization, the main focus is placed on finding the global optimum or global Pareto-optimal solutions, representing the best possible objective values. However, in practice, users may not always be interested in finding the so-called global best solutions, particularly when these solutions are quite sensitive to the variable perturbations which cannot be avoided in practice. In such cases, practitioners are interested in finding the robust solutions which are less sensitive to small perturbations in variables. Although robust optimization is dealt with in detail in single-objective evolutionary optimization studies, in this paper, we present two different robust multi-objective optimization procedures, where the emphasis is to find a robust frontier, instead of the global Pareto-optimal frontier in a problem. The first procedure is a straightforward extension of a technique used for single-objective optimization and the second procedure is a more practical approach enabling a user to set the extent of robustness desired in a problem. To demonstrate the differences between global and robust multi-objective optimization principles and the differences between the two robust optimization procedures suggested here, we develop a number of constrained and unconstrained test problems having two and three objectives and show simulation results using an evolutionary multi-objective optimization (EMO) algorithm. Finally, we also apply both robust optimization methodologies to an engineering design problem.     

Risk minimization in biometric sensor networks: an evolutionary multi-objective optimization approach

Biometric systems aim at identifying humans by their characteristics or traits. This article addresses the problem of designing a biometric sensor management unit by optimizing the risk, which is modeled as a multi-objective optimization (MO) problem with global false acceptance rate and global false rejection rate as the two objectives. In practice, when multiple biometric sensors are used, the decision is taken locally at each sensor and the data are passed to the sensor manager. At the sensor manager, the data are fused using a fusion rule and the final decision is taken. The optimization process involves designing the data fusion rule and setting of the sensor thresholds. In this work, we employ a fuzzy dominance and decomposition-based multi-objective evolutionary algorithm (MOEA) called MOEA/DFD and compare its performance with two state-of-the-art MO algorithms: MOEA/D and NSGA-II in context to the risk minimization task. 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. The MO algorithms are simulated on different number of sensor setups consisting of three, six, and eight sensors. The a priori probability of imposter is also varied from 0.1 to 0.9 to verify the performance of the system with varying degrees of threat. One of the most significant advantages of using the MO framework is that with a single run, just by changing the decision-making logic applied to the obtained Pareto front, one can find the required threshold and decision strategies for varying threats of imposter. However, with single-objective optimization, one needs to run the algorithms each time with change in the threat of imposter. Thus, multi-objective formulation of the problem appears to be more useful and better than the single-objective one. In all the test instances, MOEA/DFD performs better than all the other algorithms.     

An iterated local search for the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times

Due to its simplicity yet powerful search ability, iterated local search (ILS) has been widely used to tackle a variety of single-objective combinatorial optimization problems. However, applying ILS to solve multi-objective combinatorial optimization problems is scanty. In this paper we design a multi-objective ILS (MOILS) to solve the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times to minimize the makespan and total weighted tardiness of all jobs. In the MOILS, we design a Pareto-based variable depth search in the multi-objective local search phase. The search depth is dynamically adjusted during the search process of the MOILS to strike a balance between exploration and exploitation. We incorporate an external archive into the MOILS to store the non-dominated solutions and provide initial search points for the MOILS to escape from local optima traps. We compare the MOILS with several multi-objective evolutionary algorithms (MOEAs) shown to be effective for treating the multi-objective permutation flowshop scheduling problem in the literature. The computational results show that the proposed MOILS outperforms the MOEAs.     

高维多目标进化算法研究综述

传统的多目标进化算法能够有效地解决2个或3个目标的优化问题,但当优化目标超过4维即具有高维目标时,其优化效果将大大下降,因此高维多目标进化算法的研究得到了较多的关注.鉴于此,对高维多目标进化算法的研究进展进行系统地分类综述,分析了高维目标对优化算法造成的困难以及改进的可视化技术;总结了各类算法的特点与缺陷,并给出进一步可能的研究方向.     

基于遗传算法的转子叶片优化排序

航空发动机转子叶片按照静质量矩和频率进行优化排序对于减小叶片安装造成的不平衡量以及防止发动机颤振具有重要的意义。遗传算法是求解这类排序问题的有效方法。某型航空发动机压气机叶片排序问题可视为一个无约束多目标优化问题,为此根据目标优化的优先级将无约束多目标优化问题转化为约束单目标优化问题,建立起了叶片排序的数学模型。在此基础上,采用罚函数法进行了适应度函数设计。采用局部寻优算子来加强局部搜索能力,加速其全局优化的收敛性。大量实际数据计算结果表明,方法具有排序结果稳定、效果满意的优点,达到了对叶片优化排序的目的。     

A Multi-Objective Integer Melody Search Algorithm

There are a number of algorithms for the solution of continuous optimization problems. However, many practical design optimization problems use integer design variables instead of continuous. These types of problems cannot be handled by using continuous design variables-based algorithms. In this paper, we present a multi-objective integer melody search optimization algorithm (MO-IMS) for solving multi-objective integer optimization problems, which take design variables as integers. The proposed algorithm is a modified version of single-objective melody search (MS) algorithm, which is an innovative optimization algorithm, inspired by basic concepts applied in harmony search (HS) algorithm. Results show that MO-IMS has better performance in solving multi-objective integer problems than the existing multi-objective integer harmony search algorithm (MO-IHS). Performance of proposed algorithm is evaluated by using various performance metrics on test functions. The simulation results show that the proposed MO-IMS can be a better technique for solving multi-objective problems having integer decision variables.     

Multimodal optimization using a bi-objective evolutionary algorithm

In a multimodal optimization task, the main purpose is to find multiple optimal solutions (global and local), so that the user can have better knowledge about different optimal solutions in the search space and as and when needed, the current solution may be switched to another suitable optimum solution. To this end, evolutionary optimization algorithms (EA) stand as viable methodologies mainly due to their ability to find and capture multiple solutions within a population in a single simulation run. With the preselection method suggested in 1970, there has been a steady suggestion of new algorithms. Most of these methodologies employed a niching scheme in an existing single-objective evolutionary algorithm framework so that similar solutions in a population are deemphasized in order to focus and maintain multiple distant yet near-optimal solutions. In this paper, we use a completely different strategy in which the single-objective multimodal optimization problem is converted into a suitable bi-objective optimization problem so that all optimal solutions become members of the resulting weak Pareto-optimal set. With the modified definitions of domination and different formulations of an artificially created additional objective function, we present successful results on problems with as large as 500 optima. Most past multimodal EA studies considered problems having only a few variables. In this paper, we have solved up to 16-variable test problems having as many as 48 optimal solutions and for the first time suggested multimodal constrained test problems which are scalable in terms of number of optima, constraints, and variables. The concept of using bi-objective optimization for solving single-objective multimodal optimization problems seems novel and interesting, and more importantly opens up further avenues for research and application.     

面向复杂超多目标优化问题的自适应增强学习进化算法

在解决超多目标优化问题中,基于分解的进化算法是一种较为有效的方法.传统的分解方法依赖于一组均匀分布的参考向量,它借助聚合函数将多目标优化问题分解为一组单目标子问题,然后对这些子问题同时进行优化.然而,由于参考向量分布和Pareto前沿形状的不一致性,导致这些预定义的参考向量在解决复杂超多目标优化问题时表现较差.对此,提出一种基于自适应增强学习的超多目标进化算法(MaOEA-ABL).该算法主要分为两个阶段:第1阶段,采用一种自适应增强学习算法对预定义的参考向量进行调整,在学习过程中删除无用向量,增加新的向量;第2阶段,设计一种对Pareto形状无偏好的分解方法.为验证所提出算法的有效性,选取具有复杂Pareto前沿的MaF系列测试函数进行仿真研究,结果显示,MaOEA-ABL算法的IGD(inverted generational distance)均值在67%的测试函数上超过了对比算法,从而表明该算法在复杂超多目标优化问题中表现良好.     

Dynamic multi-objective evolutionary algorithms for single-objective optimization

This paper proposes a new method for handling the difficulty of multi-modality for the single-objective optimization problem (SOP). The method converts a SOP to an equivalent dynamic multi-objective optimization problem (DMOP). A new dynamic multi-objective evolutionary algorithm (DMOEA) is implemented to solve the DMOP. The DMOP has two objectives: the original objective and a niche-count objective. The second objective aims to maintain the population diversity for handling the multi-modality difficulty during the search process. Experimental results show that the performance of the proposed algorithm is significantly better than the state-of-the-art competitors on a set of benchmark problems and real world antenna array problems.     

Covariance matrix adaptation for multi-objective optimization

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.     

基于多点非均匀变异的多目标极值优化算法研究

多目标进化算法因其在解决含有多个矛盾目标函数的多目标优化问题中的强大处理能力,正受到越来越多的关注与研究。极值优化作为一种新型的进化算法,已在各种离散优化、连续优化测试函数以及工程优化问题中得到了较为成功的应用,但有关多目标EO算法的研究却十分有限。本文将采用Pareto优化的基本原理引入到极值优化算法中,提出一种求解连续多目标优化问题的基于多点非均匀变异的多目标极值优化算法。通过对六个国际公认的连续多目标优化测试函数的仿真实验结果表明：本文提出算法相比NSGA-II、 PAES、SPEA和SPEA2等经典多目标优化算法在收敛性和分布性方面均具有优势。     

On redundant coverage maximization in wireless visual sensor networks: Evolutionary algorithms for multi-objective optimization

Wireless visual sensor networks can provide valuable information for a variety of monitoring and control applications. Frequently, a set of targets must be covered by visual sensors, as such visual sensing redundancy is a desired condition specially when applications have availability requirements for multiple coverage perspectives. If visual sensors become rotatable, their sensing orientations can be adjusted to optimize coverage and redundancy, bringing different challenges as there may be different coverage optimization objectives. Actually, the specific issue of redundant coverage maximization is inherently a multi-objective problem, but usual approaches are not designed accordingly to compute visual sensing redundancy. This article proposes two different evolutionary algorithms that exploit the multi-objective nature of the redundant coverage maximization problem: a lexicographic ”a priori” algorithm and a NSGA-II ”a posteriori” algorithm. The performance of both algorithms are compared, using a previously proposed single-objective greedy-based algorithm as a reference. Numerical results outline the benefits of employing evolutionary algorithms for adjustments of sensors’ orientations, potentially benefiting deployment and management of wireless visual sensor networks for different monitoring scenarios.     

Multi-center variable-scale search algorithm for combinatorial optimization problems with the multimodal property

Combinatorial optimization problems (COPs) are discrete problems arising from aerospace, bioinformatics, manufacturing, and other fields. One of the classic COPs is the scheduling problem. Moreover, these problems are usually multimodal optimization problems with a quantity of global and local optima. As a result, many search algorithms can easily become trapped into local optima. In this article, we propose a multi-center variable-scale search algorithm for solving both single-objective and multi-objective COPs. The algorithm consists of two distinct points. First, the multi-center strategy chooses several individuals with better performance as the only parents of the next generation, which means that there are a number of separate searching areas around the searching center. Second, the next generation of the population is produced by a variable-scale strategy with an exponential equation based on the searching center. The equation is designed to control the neighborhood scale, and adaptively realize the large-scale and small-scale searches at different search stages to balance the maintenance of diversity and convergence speed. In addition, an approach of adjusting centers is proposed concerning the number and distribution of centers for solving multi-objective COPs. Finally, the proposed algorithm is applied to three COPs, including the well-known flexible job shop scheduling problem, the unrelated parallel machine scheduling problem, and the test task scheduling problem. Both the single-objective optimization algorithm and the multi-objective optimization algorithm demonstrate competitive performance compared with existing methods.