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A Fast and Elitist MultiObjective Genetic Algorithm: NSGAII
, 2000
"... Multiobjective evolutionary algorithms which use nondominated sorting and sharing have been mainly criticized for their (i) O(MN computational complexity (where M is the number of objectives and N is the population size), (ii) nonelitism approach, and (iii) the need for specifying a sharing param ..."
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Cited by 966 (41 self)
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Multiobjective evolutionary algorithms which use nondominated sorting and sharing have been mainly criticized for their (i) O(MN computational complexity (where M is the number of objectives and N is the population size), (ii) nonelitism approach, and (iii) the need for specifying a sharing parameter. In this paper, we suggest a nondominated sorting based multiobjective evolutionary algorithm (we called it the Nondominated Sorting GAII or NSGAII) which alleviates all the above three difficulties. Specifically, a fast nondominated sorting approach with O(MN ) computational complexity is presented. Second, a selection operator is presented which creates a mating pool by combining the parent and child populations and selecting the best (with respect to fitness and spread) N solutions. Simulation results on a number of difficult test problems show that the proposed NSGAII, in most problems, is able to find much better spread of solutions and better convergence near the true Paretooptimal front compared to PAES and SPEA  two other elitist multiobjective EAs which pay special attention towards creating a diverse Paretooptimal front. Moreover, we modify the definition of dominance in order to solve constrained multiobjective problems eciently. Simulation results of the constrained NSGAII on a number of test problems, including a fiveobjective, sevenconstraint nonlinear problem, are compared with another constrained multiobjective optimizer and much better performance of NSGAII is observed. Because of NSGAII's low computational requirements, elitist approach, parameterless niching approach, and simple constrainthandling strategy, NSGAII should find increasing applications in the coming years.
Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach
 IEEE Transactions on Evolutionary Computation
, 1999
"... Abstract—Evolutionary algorithms (EA’s) are often wellsuited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in ..."
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Cited by 577 (19 self)
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Abstract—Evolutionary algorithms (EA’s) are often wellsuited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EA’s are compared quantitatively where an extended 0/1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the Strength Pareto EA (SPEA), that combines several features of previous multiobjective EA’s in a unique manner. It is characterized by a) storing nondominated solutions externally in a second, continuously updated population, b) evaluating an individual’s fitness dependent on the number of external nondominated points that dominate it, c) preserving population diversity using the Pareto dominance relationship, and d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proofofprinciple results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware–software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Paretooptimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EA’s on the 0/1 knapsack problem. Index Terms — Clustering, evolutionary algorithm, knapsack problem, multiobjective optimization, niching, Pareto optimality.
Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications
, 1999
"... Many realworld problems involve two types of problem difficulty: i) multiple, conflicting objectives and ii) a highly complex search space. On the one hand, instead of a single optimal solution competing goals give rise to a set of compromise solutions, generally denoted as Paretooptimal. In the a ..."
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Cited by 340 (13 self)
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Many realworld problems involve two types of problem difficulty: i) multiple, conflicting objectives and ii) a highly complex search space. On the one hand, instead of a single optimal solution competing goals give rise to a set of compromise solutions, generally denoted as Paretooptimal. In the absence of preference information, none of the corresponding tradeoffs can be said to be better than the others. On the other hand, the search space can be too large and too complex to be solved by exact methods. Thus, efficient optimization strategies are required that are able to deal with both difficulties. Evolutionary algorithms possess several characteristics that are desirable for this kind of problem and make them preferable to classical optimization methods. In fact, various evolutionary approaches to multiobjective optimization have been proposed since 1985, capable of searching for multiple Paretooptimal solutions concurrently in a single simulation run. However, in spite of this...
Multiobjective Evolutionary Algorithms: Analyzing the StateoftheArt
, 2000
"... Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mideighties in an attempt to stochastically solve problems of this generic class. During the past decade, ..."
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Cited by 326 (7 self)
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Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mideighties in an attempt to stochastically solve problems of this generic class. During the past decade, a variety of multiobjective EA (MOEA) techniques have been proposed and applied to many scientific and engineering applications. Our discussion's intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs. Current MOEA theoretical developments are evaluated; specific topics addressed include fitness functions, Pareto ranking, niching, fitness sharing, mating restriction, and secondary populations. Since the development and application of MOEAs is a dynamic and rapidly growing activity, we focus on key analytical insights based upon critical MOEA evaluation of c...
A Comprehensive Survey of EvolutionaryBased Multiobjective Optimization Techniques
 Knowledge and Information Systems
, 1998
"... . This paper presents a critical review of the most important evolutionarybased multiobjective optimization techniques developed over the years, emphasizing the importance of analyzing their Operations Research roots as a way to motivate the development of new approaches that exploit the search cap ..."
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Cited by 240 (21 self)
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. This paper presents a critical review of the most important evolutionarybased multiobjective optimization techniques developed over the years, emphasizing the importance of analyzing their Operations Research roots as a way to motivate the development of new approaches that exploit the search capabilities of evolutionary algorithms. Each technique is briefly described mentioning its advantages and disadvantages, their degree of applicability and some of their known applications. Finally, the future trends in this discipline and some of the open areas of research are also addressed. Keywords: multiobjective optimization, multicriteria optimization, vector optimization, genetic algorithms, evolutionary algorithms, artificial intelligence. 1 Introduction Since the pioneer work of Rosenberg in the late 60s regarding the possibility of using geneticbased search to deal with multiple objectives, this new area of research (now called evolutionary multiobjective optimization) has grown c...
MultiObjective Genetic Algorithms: Problem Difficulties and Construction of Test Problems
 Evolutionary Computation
, 1999
"... In this paper, we study the problem features that may cause a multiobjective genetic algorithm (GA) difficulty in converging to the true Paretooptimal front. Identification of such features helps us develop difficult test problems for multiobjective optimization. Multiobjective test problems ..."
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Cited by 167 (12 self)
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In this paper, we study the problem features that may cause a multiobjective genetic algorithm (GA) difficulty in converging to the true Paretooptimal front. Identification of such features helps us develop difficult test problems for multiobjective optimization. Multiobjective test problems are constructed from singleobjective optimization problems, thereby allowing known difficult features of singleobjective problems (such as multimodality, isolation, or deception) to be directly transferred to the corresponding multiobjective problem. In addition, test problems having features specific to multiobjective optimization are also constructed. More importantly, these difficult test problems will enable researchers to test their algorithms for specific aspects of multiobjective optimization. Keywords Genetic algorithms, multiobjective optimization, niching, paretooptimality, problem difficulties, test problems. 1 Introduction After a decade since the pioneering wor...
Covariance Matrix Adaptation for Multiobjective Optimization
 Evolutionary Computation
"... The covariance matrix adaptation evolution strategy (CMAES) is one of the most powerful evolutionary algorithms for realvalued singleobjective optimization. In this paper, we develop a variant of the CMAES for multiobjective optimization (MOO). We first introduce a singleobjective, elitist C ..."
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Cited by 57 (8 self)
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The covariance matrix adaptation evolution strategy (CMAES) is one of the most powerful evolutionary algorithms for realvalued singleobjective optimization. In this paper, we develop a variant of the CMAES for multiobjective optimization (MOO). We first introduce a singleobjective, elitist CMAES using plusselection and step size control based on a success rule. This algorithm is compared to the standard CMAES. The elitist CMAES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in suboptimal local minima. In the new multiobjective CMAES (MOCMAES) a population of individuals that adapt their search strategy as in the elitist CMAES is maintained. These are subject to multiobjective selection. The selection is based on nondominated sorting using either the crowdingdistance or the contributing hypervolume as second sorting criterion. Both the elitist singleobjective CMAES and the MOCMAES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMAES. The benefits of the new MOCMAES in comparison to the wellknown NSGAII and to NSDE, a multiobjective differential evolution algorithm, are experimentally shown.
An EMO Algorithm Using the Hypervolume Measure as Selection Criterion
 2005 Int’l Conference, March 2005
, 2005
"... The hypervolume measure is one of the most frequently applied measures for comparing the results of evolutionary multiobjective optimization algorithms (EMOA). The idea to use this measure for selection is selfevident. A steadystate EMOA will be devised, that combines concepts of nondominated sor ..."
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Cited by 45 (8 self)
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The hypervolume measure is one of the most frequently applied measures for comparing the results of evolutionary multiobjective optimization algorithms (EMOA). The idea to use this measure for selection is selfevident. A steadystate EMOA will be devised, that combines concepts of nondominated sorting with a selection operator based on the hypervolume measure. The algorithm computes a well distributed set of solutions with bounded size thereby focussing on interesting regions of the Pareto front(s). By means of standard benchmark problems the algorithm will be compared to other well established EMOA. The results show that our new algorithm achieves good convergence to the Pareto front and outperforms standard methods in the hypervolume covered.
Evolutionary Algorithms for MultiCriterion Optimization in Engineering Design
, 1999
"... this paper, we briefly outline the principles of multiobjective optimization. Thereafter, we discuss why classical search and optimization methods are not adequate for multicriterion optimization by discussing the working of two popular methods. We then outline several evolutionary methods for han ..."
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Cited by 43 (0 self)
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this paper, we briefly outline the principles of multiobjective optimization. Thereafter, we discuss why classical search and optimization methods are not adequate for multicriterion optimization by discussing the working of two popular methods. We then outline several evolutionary methods for handling multicriterion optimization problems. Of them, we discuss one implementation (nondominated sorting GA or NSGA [38]) in somewhat greater details. Thereafter, we demonstrate the working of the evolutionary methods by applying NSGA to three test problems having constraints and discontinuous Paretooptimal region. We also show the efficacy of evolutionary algorithms in engineering design problems by solving a welded beam design problem. The results show that evolutionary methods can find widely different yet nearParetooptimal solutions in such problems. Based on the above studies, this paper also suggests a number of immediate future studies which would make this emerging field more mature and applicable in practice. 1.2 PRINCIPLES OF MULTICRITERION OPTIMIZATION
The Measure of Pareto Optima. Applications to Multiobjective Metaheuristics
 Evolutionary MultiCriterion Optimization. Second International Conference, EMO 2003
, 2003
"... Abstract. This article describes a set function that maps a set of Pareto optimal points to a scalar. A proof is presented that shows that the maximization of this scalar value constitutes the necessary and sufficient condition for the function’s arguments to be maximally diverse Pareto optimal solu ..."
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Cited by 36 (3 self)
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Abstract. This article describes a set function that maps a set of Pareto optimal points to a scalar. A proof is presented that shows that the maximization of this scalar value constitutes the necessary and sufficient condition for the function’s arguments to be maximally diverse Pareto optimal solutions of a discrete, multiobjective, optimization problem. This scalar quantity, a hypervolume based on a Lebesgue measure, is therefore the best metric to assess the quality of multiobjective optimization algorithms. Moreover, it can be used as the objective function in simulated annealing (SA) to induce convergence in probability to the Pareto optima. An efficient algorithm for calculating this scalar and analysis of its complexity is presented. 1