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Showing papers on "Multi-objective optimization published in 2000"


Journal ArticleDOI
TL;DR: This paper provides a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions and shows that elitism is shown to be an important factor for improving evolutionary multiobjectives search.
Abstract: In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in converging to the Pareto-optimal front (e.g., multimodality and deception). By investigating these different problem features separately, it is possible to predict the kind of problems to which a certain technique is or is not well suited. However, in contrast to what was suspected beforehand, the experimental results indicate a hierarchy of the algorithms under consideration. Furthermore, the emerging effects are evidence that the suggested test functions provide sufficient complexity to compare multiobjective optimizers. Finally, elitism is shown to be an important factor for improving evolutionary multiobjective search.

4,867 citations


Journal ArticleDOI
TL;DR: The Pareto Archived Evolution Strategy (PAES) as discussed by the authors is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors.
Abstract: We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES) We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods We introduce (1 + λ) and (μ | λ) variants of PAES as extensions to the basic algorithm Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions This allows standard statistical analysis to be carried out for comparative purposes Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks

2,140 citations


Journal ArticleDOI
TL;DR: A novel approach to balance objective and penalty functions stochastically, i.e., stochastic ranking, is introduced, and a new view on penalty function methods in terms of the dominance of penalty and objective functions is presented.
Abstract: Penalty functions are often used in constrained optimization. However, it is very difficult to strike the right balance between objective and penalty functions. This paper introduces a novel approach to balance objective and penalty functions stochastically, i.e., stochastic ranking, and presents a new view on penalty function methods in terms of the dominance of penalty and objective functions. Some of the pitfalls of naive penalty methods are discussed in these terms. The new ranking method is tested using a (/spl mu/, /spl lambda/) evolution strategy on 13 benchmark problems. Our results show that suitable ranking alone (i.e., selection), without the introduction of complicated and specialized variation operators, is capable of improving the search performance significantly.

1,571 citations


Journal ArticleDOI
TL;DR: The intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs.
Abstract: Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mid-eighties 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 current research and applications. Recommended MOEA designs are presented, along with conclusions and recommendations for future work.

1,241 citations


Journal ArticleDOI
TL;DR: The notion of using co-evolution to adapt the penalty factors of a fitness function incorporated in a genetic algorithm (GA) for numerical optimization is introduced.

1,096 citations


Book ChapterDOI
18 Sep 2000
TL;DR: This work introduces a new multiobjective evolutionary algorithm called PESA (the Pareto Envelope-based Selection Algorithm), in which selection and diversity maintenance are controlled via a simple hyper-grid based scheme.
Abstract: We introduce a new multiobjective evolutionary algorithm called PESA (the Pareto Envelope-based Selection Algorithm), in which selection and diversity maintenance are controlled via a simple hyper-grid based scheme. PESA's selection method is relatively unusual in comparison with current well known multiobjective evolutionary algorithms, which tend to use counts based on the degree to which solutions dominate others in the population. The diversity maintenance method is similar to that used by certain other methods. The main attraction of PESA is the integration of selection and diversity maintenance, whereby essentially the same technique is used for both tasks. The resulting algorithm is simple to describe, with full pseudocode provided here and real code available from the authors. We compare PESA with two recent strong-performing MOEAs on some multiobjective test problems recently proposed by Deb. We find that PESA emerges as the best method overall on these problems.

905 citations


01 Jan 2000
TL;DR: The results of the automated optimization cycle show shapes previously obtained by physical understanding as well as novel shapes of even higher eciency.
Abstract: Multiobjective evolutionary algorithms for shape optimization of electrokinetic micro channels have been developed and implemented. An extension to the Strength Pareto Approach that enables targeting has been developed. The results of the automated optimization cycle show shapes previously obtained by physical understanding as well as novel shapes of even higher eciency.

786 citations


Journal ArticleDOI
TL;DR: The purpose of this paper is to summarize and organize the information on current evolutionary-based approaches, emphasizing the importance of analyzing the operations research techniques in which most of them are based, in an attempt to motivate researchers to look into these mathematical programming approaches for new ways of exploiting the search capabilities of evolutionary algorithms.
Abstract: After using evolutionary techniques for single-objective optimization during more than two decades, the incorporation of more than one objective in the fitness function has finally become a popular area of research. As a consequence, many new evolutionary-based approaches and variations of existing techniques have recently been published in the technical literature. The purpose of this paper is to summarize and organize the information on these current approaches, emphasizing the importance of analyzing the operations research techniques in which most of them are based, in an attempt to motivate researchers to look into these mathematical programming approaches for new ways of exploiting the search capabilities of evolutionary algorithms. Furthermore, a summary of the main algorithms behind these approaches is provided, together with a brief criticism that includes their advantages and disadvantages, degree of applicability, and some known applications. Finally, further trends in this area and some possible paths for further research are also addressed.

762 citations


Journal ArticleDOI
TL;DR: An overview of the methods that have been developed since 1977 for solving various reliability optimization problems; applications of these methods to various types of design problems; and heuristics, metaheuristic algorithms, exact methods, reliability-redundancy allocation, multi-objective optimization and assignment of interchangeable components in reliability systems.
Abstract: This paper provides: an overview of the methods that have been developed since 1977 for solving various reliability optimization problems; applications of these methods to various types of design problems; and heuristics, metaheuristic algorithms, exact methods, reliability-redundancy allocation, multi-objective optimization and assignment of interchangeable components in reliability systems. Like other applications, exact solutions for reliability optimization problems are not necessarily desirable because exact solutions are difficult to obtain, and even when they are available, their utility is marginal. A majority of the work in this area is devoted to developing heuristic and metaheuristic algorithms for solving optimal redundancy-allocation problems.

636 citations


Journal ArticleDOI
TL;DR: A steepest descent method for unconstrained multicriteria optimization and a “feasible descent direction” method for the constrained case, both of which converge to a point satisfying certain first-order necessary conditions for Pareto optimality.
Abstract: We propose a steepest descent method for unconstrained multicriteria optimization and a “feasible descent direction” method for the constrained case. In the unconstrained case, the objective functions are assumed to be continuously differentiable. In the constrained case, objective and constraint functions are assumed to be Lipshitz-continuously differentiable and a constraint qualification is assumed. Under these conditions, it is shown that these methods converge to a point satisfying certain first-order necessary conditions for Pareto optimality. Both methods do not scalarize the original vector optimization problem. Neither ordering information nor weighting factors for the different objective functions are assumed to be known. In the single objective case, we retrieve the Steepest descent method and Zoutendijk's method of feasible directions, respectively.

497 citations


Proceedings ArticleDOI
12 Nov 2000
TL;DR: In this paper, a polynomially succinct curve that approximates the Pareto curve for any /spl epsiv/approximation is presented, and a necessary and sufficient condition under which such a curve can be constructed in time polynomial in the size of the instance and 1/spl eπiv/
Abstract: We study problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives (the so-called Pareto curve). We point out that, under very general conditions, there is a polynomially succinct curve that /spl epsiv/-approximates the Pareto curve, for any /spl epsiv/>0. We give a necessary and sufficient condition under which this approximate Pareto curve can be constructed in time polynomial in the size of the instance and 1//spl epsiv/. In the case of multiple linear objectives, we distinguish between two cases: when the underlying feasible region is convex, then we show that approximating the multi-objective problem is equivalent to approximating the single-objective problem. If however the feasible region is discrete, then we point out that the question reduces to an old and recurrent one: how does the complexity of a combinatorial optimization problem change when its feasible region is intersected with a hyperplane with small coefficients; we report some interesting new findings in this domain. Finally, we apply these concepts and techniques to formulate and solve approximately a cost-time-quality trade-off for optimizing access to the World-Wide Web, in a model first studied by Etzioni et al. (1996) (which was actually the original motivation for this work).

Journal ArticleDOI
TL;DR: A new optimization principle is presented that is particularly suited for more complex optimization problems (“discontinuous” ones, problems with hard-to-find admissible solutions, Problems with complex objectives or many constraints).

Proceedings ArticleDOI
16 Jul 2000
TL;DR: A memetic algorithm for tackling multiobjective optimization problems is presented that employs the proven local search strategy used in the Pareto archived evolution strategy (PAES) and combines it with the use of a population and recombination.
Abstract: A memetic algorithm for tackling multiobjective optimization problems is presented. The algorithm employs the proven local search strategy used in the Pareto archived evolution strategy (PAES) and combines it with the use of a population and recombination. Verification of the new M-PAES (memetic PAES) algorithm is carried out by testing it on a set of multiobjective 0/1 knapsack problems. On each problem instance, a comparison is made between the new memetic algorithm, the (1+1)-PAES local searcher, and the strength Pareto evolutionary algorithm (SPEA) of E. Zitzler and L. Thiele (1998, 1999).

Journal ArticleDOI
TL;DR: In this article, trade-off surfaces give a way of visualizing the alternative compromises, and value functions (or utility functions) identify the part of the surface on which optimal solutions lie.

Journal ArticleDOI
TL;DR: The proposed technique does not require fine tuning of factors like the traditional penalty function and uses a self-adaptation mechanism that avoids the traditional empirical adjustment of the main genetic operators (i.e., crossover and mutation).
Abstract: In this paper, we introduce the concept of non-dominance (commonly used in multi-objective optimization) as a way to incorporate constraints into the fitness function of a genetic algorithm Each individual is assigned a rank based on its degree of dominance over the rest of the population Feasible individuals are always ranked higher than infeasible ones, and the degree of constraint violation determines the rank among infeasible individuals The proposed technique does not require fine tuning of factors like the traditional penalty function and uses a self-adaptation mechanism that avoids the traditional empirical adjustment of the main genetic operators (ie, crossover and mutation)

Journal ArticleDOI
TL;DR: The general background of this area is presented, followed by a description of how the results can be described in terms of Pareto sets, and the several methods available for generating optimal solutions.
Abstract: Multiobjective optimization involves the simultaneous optimization of more than one objective function. This is quite commonly encountered in Chemical Engineering. A considerable amount of research has been reported in this area over the last twenty years. These are reviewed in the present paper. The general background of this area is presented at the beginning, followed by a description of how the results can be described in terms of Pareto sets. We then present the several methods available for generating these optimal solutions. Applications of optimization in Chemical Engineering wherein multiple objectives are encountered, as well as special adaptations of the basic algorithms required to solve these problems, are then discussed. Some comments are also made on possible directions that future research may take in this area.

Proceedings ArticleDOI
16 Jul 2000
TL;DR: The most important preference handling approaches used with evolutionary algorithms, analyzing their advantages and disadvantages are reviewed, and then, some of the potential areas of future research in this discipline are proposed.
Abstract: Despite the relatively high volume of research conducted on evolutionary multiobjective optimization in the last few years. Little attention has been paid to the decision making process that is required to select a final solution to the multiobjective optimization problem at hand. This paper reviews the most important preference handling approaches used with evolutionary algorithms, analyzing their advantages and disadvantages, and then, it proposes some of the potential areas of future research in this discipline.

Journal ArticleDOI
10 Sep 2000
TL;DR: Several new set quality metrics are introduced that can be used to assess the ‘‘goodness’’ of an observed Pareto solution set, which should enable a designer to either monitor the quality of an observation P solution set as obtained by a multiobjective optimization method, or compare the sets as reported by different multiObjective optimization methods.
Abstract: In this paper, several new set quality metrics are introduced that can be used to eva the ‘‘goodness’’ of an observed Pareto solution set. These metrics, which are formu in closed-form and geometrically illustrated, include hyperarea difference, Pareto spr accuracy of an observed Pareto frontier, number of distinct choices and cluster. metrics should enable a designer to either monitor the quality of an observed P solution set as obtained by a multiobjective optimization method, or compare the qu of observed Pareto solution sets as reported by different multiobjective optimization m ods. A vibrating platform example is used to demonstrate the calculation of these m for an observed Pareto solution set. @DOI: 10.1115/1.1329875 #

Journal ArticleDOI
TL;DR: The results obtained show that the new approach to handle constraints using evolutionary algorithms can consistently outperform the other techniques using relatively small sub-populations, and without a significant sacrifice in terms of performance.
Abstract: This paper presents a new approach to handle constraints using evolutionary algorithms. The new technique treats constraints as objectives, and uses a multiobjective optimization approach to solve the re-stated single-objective optimization problem. The new approach is compared against other numerical and evolutionary optimization techniques in several engineering optimization problems with different kinds of constraints. The results obtained show that the new approach can consistently outperform the other techniques using relatively small sub-populations, and without a significant sacrifice in terms of performance.

Journal ArticleDOI
TL;DR: Using the concept of min–max optimum, a new GA-based multiobjective optimization technique is proposed and two truss design problems are solved using it, proving that this technique generates better trade-offs and that the genetic algorithm can be used as a reliable numerical optimization tool.

Proceedings ArticleDOI
16 Jul 2000
TL;DR: Four abstract evolutionary algorithms for multi-objective optimization and theoretical results that characterize their convergence behavior are presented and it is easy to verify whether or not a particular instantiation of these abstract evolutionary algorithm offers the desired limit behavior.
Abstract: We present four abstract evolutionary algorithms for multi-objective optimization and theoretical results that characterize their convergence behavior. Thanks to these results it is easy to verify whether or not a particular instantiation of these abstract evolutionary algorithms offers the desired limit behavior. Several examples are given.

Journal ArticleDOI
TL;DR: This paper describes a novel implementation of the Simulated Annealing algorithm designed to explore the trade-off between multiple objectives in optimization problems and concludes that the proposed algorithm offers an effective and easily implemented method for exploring thetrade-off in multiobjective optimization problems.
Abstract: This paper describes a novel implementation of the Simulated Annealing algorithm designed to explore the trade-off between multiple objectives in optimization problems. During search, the algorithm maintains and updates an archive of non-dominated solutions between each of the competing objectives. At the end of search, the final archive corresponds to a number of optimal solutions from which the designer may choose a particular configuration. A new acceptance probability formulation based on an annealing schedule with multiple temperatures (one for each objective) is proposed along with a novel restart strategy. The performance of the algorithm is demonstrated on three examples. It is concluded that the proposed algorithm offers an effective and easily implemented method for exploring the trade-off in multiobjective optimization problems.

Journal ArticleDOI
TL;DR: Some recent results and current research trends on deterministic and stochastic global optimization and global continuous approaches to discrete optimization are highlighted.

Journal ArticleDOI
TL;DR: It is shown that the approach proposed can give a compromise solution that is not only non-dominated but also optimal in a certain sense and robust enough to cover a wide spectrum of methods.

Journal ArticleDOI
TL;DR: The results show that the implementation of the RBF algorithm is very efficient on the standard test problems compared to other known solvers, but even more interesting, it performs extremely well on the train design optimization problem.
Abstract: The paper considers global optimization of costly objective functions, i.e. the problem of finding the global minimum when there are several local minima and each function value takes considerable CPU time to compute. Such problems often arise in industrial and financial applications, where a function value could be a result of a time-consuming computer simulation or optimization. Derivatives are most often hard to obtain, and the algorithms presented make no use of such information. Several algorithms to handle the global optimization problem are described, but the emphasis is on a new method by Gutmann and Powell, A radial basis function method for global optimization. This method is a response surface method, similar to the Efficient Global Optimization (EGO) method of Jones. Our Matlab implementation of the Radial Basis Function (RBF) method is described in detail and we analyze its efficiency on the standard test problem set of Dixon-Szego, as well as its applicability on a real life industrial problem from train design optimization. The results show that our implementation of the RBF algorithm is very efficient on the standard test problems compared to other known solvers, but even more interesting, it performs extremely well on the train design optimization problem.

Journal ArticleDOI
TL;DR: The use of response surface estimation in collaborative optimization, an architecture for large-scale multidisciplinary design is described, and how response surface models of subproblem optimization results improve the performance of collaborative optimization is demonstrated.
Abstract: The use of response surface estimation in collaborative optimization, an architecture for large-scale multidisciplinary design is described. Collaborative optimization preserves the autonomy of individual disciplines while providing a mechanism for coordinating the overall design problem and progressing toward improved designs. Collaborative optimization is a two-level optimization architecture, with discipline-specific optimizations free to specify local designs, and a global optimization that ensures that all of the discipline designs eventually agree on a single value for those variables that are shared in common. Results demonstrate how response surface models of subproblem optimization results improve the performance of collaborative optimization. The utility of response surface estimation in collaborative optimization depends on the generation of inexpensive accurate response surface models and the refinement of these models over several fitting cycles. Special properties of the subproblem optimization formulation are exploited to reduce the number of required subproblem optimizations to develop a quadratic model from O(n 2 ) to O(n/2). Response surface refinement is performed using ideas from trust region methods. Results for the combined approaches are compared through the design optimization of a tailless unmanned air vehicle in 44 design variables.

Journal ArticleDOI
TL;DR: In this paper, a genetic-algorithm-based procedure for solving multi-objective network level pavement maintenance programming problems was developed, where the concepts of Pareto optimal solution set and rank-based fitness evaluation, and two methods of selecting an optimal solution, were adopted.
Abstract: Pavement maintenance planning and programming requires optimization analysis involving multiobjective considerations. Traditionally single-objective optimization techniques have been employed by pavement researchers and practitioners because of the complexity involved in multiobjective analysis. This paper develops a genetic-algorithm-based procedure for solving multiobjective network level pavement maintenance programming problems. The concepts of Pareto optimal solution set and rank-based fitness evaluation, and two methods of selecting an optimal solution, were adopted. It was found that the robust search characteristics and multiple-solution handling capability of genetic-algorithms were well suited for multiobjective optimization analysis. Formulation and development of the solution algorithm were described and demonstrated with a numerical example problem in which a hypothetical network level pavement maintenance programming analysis was performed for two- and three-objective optimization, respectively. A comparison between the two- and three-objective solutions was made to highlight some practical considerations in applying multiobjective optimization to pavement maintenance management.


Journal ArticleDOI
TL;DR: In this paper, a fuzzy discrete multicriteria cost optimization model for design of space steel structures subjected to the actual constraints of commonly-used design codes such as the AISC ASD code is presented.
Abstract: Only a small fraction of the hundreds of papers published on optimization of steel structures deal with cost optimization; the great majority deal only with minimization of the weight of the structure. Those few that are concerned with cost optimization deal with small two-dimensional or academic examples. In this article, the writers present a fuzzy discrete multicriteria cost optimization model for design of space steel structures subjected to the actual constraints of commonly-used design codes such as the AISC ASD code by considering three design criteria: (1) minimum material cost; (2) minimum weight; and (3) minimum number of different section types. The computational model starts with a continuous-variable minimum weight solution with a preemptive constraint violation strategy as the lower bound followed by a fuzzy discrete multicriteria optimization. It is concluded that solving the structural design problem as a cost optimization problem can result in substantial cost savings compared with the tr...

Journal ArticleDOI
TL;DR: The NIMBUS algorithm and its implementation WWW-NIMBUS is described, which is the first interactive multiobjective optimization system on the Internet, and the main principles of its implementation are centralized computing and a distributed interface.