An efficient constraint handling method for genetic algorithms
09 Jun 2000-Computer Methods in Applied Mechanics and Engineering (North-Holland)-Vol. 186, Iss: 2, pp 311-338
TL;DR: GA's population-based approach and ability to make pair-wise comparison in tournament selection operator are exploited to devise a penalty function approach that does not require any penalty parameter to guide the search towards the constrained optimum.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2000-06-09. It has received 3495 citations till now. The article focuses on the topics: Penalty method & Feasible region.
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TL;DR: This paper suggests a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties, and modify the definition of dominance in order to solve constrained multi-objective problems efficiently.
Abstract: Multi-objective evolutionary algorithms (MOEAs) that use non-dominated sorting and sharing have been criticized mainly for: (1) their O(MN/sup 3/) computational complexity (where M is the number of objectives and N is the population size); (2) their non-elitism approach; and (3) the need to specify a sharing parameter. In this paper, we suggest a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN/sup 2/) computational complexity is presented. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best N solutions (with respect to fitness and spread). Simulation results on difficult test problems show that NSGA-II is able, for most problems, to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the Pareto-archived evolution strategy and the strength-Pareto evolutionary algorithm - two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems efficiently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint nonlinear problem, are compared with another constrained multi-objective optimizer, and the much better performance of NSGA-II is observed.
37,111 citations
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TL;DR: The results of the classical engineering design problems and real application prove that the proposed GWO algorithm is applicable to challenging problems with unknown search spaces.
10,082 citations
Cites methods from "An efficient constraint handling me..."
...Coello [64] and Deb [65,66] employed GA, whereas Lee and Geem [67] used HS to solve this problem....
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...Algorithm Optimum variables Optimum cost Ts Th R L GWO 0.812500 0.434500 42.089181 176.758731 6051.5639 GSA 1.125000 0.625000 55.9886598 84.4542025 8538.8359 PSO (He and Wang) 0.812500 0.437500 42.091266 176.746500 6061.0777 GA (Coello) 0.812500 0.434500 40.323900 200.000000 6288.7445 GA (Coello and Montes) 0.812500 0.437500 42.097398 176.654050 6059.9463 GA (Deb and Gene) 0.937500 0.500000 48.329000 112.679000 6410.3811 ES (Montes and Coello) 0.812500 0.437500 42.098087 176.640518 6059.7456 DE (Huang et al.)...
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...The mathematical formulation is as follows: Consider ~x ¼ ½x1 x2 x3 x4 ¼ ½hltb ; Minimize ðf~xÞ ¼ 1:10471x21x2 þ 0:04811x3x4ð14:0þ x2Þ; Subject to g1ð~xÞ ¼ sð~xÞ smax 6 0; g2ð~xÞ ¼ rð~xÞ rmax 6 0; g3ð~xÞ ¼ dð~xÞ dmax 6 0; g4ð~xÞ ¼ x1 x4 6 0; g5ð~xÞ ¼ P Pcð~xÞ 6 0; g6ð~xÞ ¼ 0:125 x1 6 0 g7ð~xÞ ¼ 1:10471x21 þ 0:04811x3x4ð14:0þ x2Þ 5:0 6 0 ð5:2Þ Variable range 0:1 6 x1 6 2; 0:1 6 x2 6 10; 0:1 6 x3 6 10; 0:1 6 x4 6 2 Coello [64] and Deb [65,66] employed GA, whereas Lee and Geem [67] used HS to solve this problem....
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TL;DR: Optimization results prove that the WOA algorithm is very competitive compared to the state-of-art meta-heuristic algorithms as well as conventional methods.
7,090 citations
Book•
30 Jun 2002
TL;DR: This paper presents a meta-anatomy of the multi-Criteria Decision Making process, which aims to provide a scaffolding for the future development of multi-criteria decision-making systems.
Abstract: List of Figures. List of Tables. Preface. Foreword. 1. Basic Concepts. 2. Evolutionary Algorithm MOP Approaches. 3. MOEA Test Suites. 4. MOEA Testing and Analysis. 5. MOEA Theory and Issues. 3. MOEA Theoretical Issues. 6. Applications. 7. MOEA Parallelization. 8. Multi-Criteria Decision Making. 9. Special Topics. 10. Epilog. Appendix A: MOEA Classification and Technique Analysis. Appendix B: MOPs in the Literature. Appendix C: Ptrue & PFtrue for Selected Numeric MOPs. Appendix D: Ptrue & PFtrue for Side-Constrained MOPs. Appendix E: MOEA Software Availability. Appendix F: MOEA-Related Information. Index. References.
5,994 citations
TL;DR: A detailed review of the basic concepts of DE and a survey of its major variants, its application to multiobjective, constrained, large scale, and uncertain optimization problems, and the theoretical studies conducted on DE so far are presented.
Abstract: Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms in current use. DE operates through similar computational steps as employed by a standard evolutionary algorithm (EA). However, unlike traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used for generating the offspring. Since its inception in 1995, DE has drawn the attention of many researchers all over the world resulting in a lot of variants of the basic algorithm with improved performance. This paper presents a detailed review of the basic concepts of DE and a survey of its major variants, its application to multiobjective, constrained, large scale, and uncertain optimization problems, and the theoretical studies conducted on DE so far. Also, it provides an overview of the significant engineering applications that have benefited from the powerful nature of DE.
4,321 citations
References
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Book•
01 Sep 1988
TL;DR: In this article, the authors present the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields, including computer programming and mathematics.
Abstract: From the Publisher:
This book brings together - in an informal and tutorial fashion - the computer techniques, mathematical tools, and research results that will enable both students and practitioners to apply genetic algorithms to problems in many fields
Major concepts are illustrated with running examples, and major algorithms are illustrated by Pascal computer programs No prior knowledge of GAs or genetics is assumed, and only a minimum of computer programming and mathematics background is required
52,797 citations
01 Jan 1989
12,457 citations
"An efficient constraint handling me..." refers methods in this paper
...Among them, niching methods [16] and use of mutation [17] are popular ones....
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05 Jul 1995
TL;DR: C Culling is near optimal for this problem, highly noise tolerant, and the best known a~~roach in some regimes, and some new large deviation bounds on this submartingale enable us to determine the running time of the algorithm.
Abstract: We analyze the performance of a Genetic Type Algorithm we call Culling and a variety of other algorithms on a problem we refer to as ASP. Culling is near optimal for this problem, highly noise tolerant, and the best known a~~roach . . in some regimes. We show that the problem of learning the Ising perception is reducible to noisy ASP. These results provide an example of a rigorous analysis of GA’s and give insight into when and how C,A’s can beat competing methods. To analyze the genetic algorithm, we view it as a special type of submartingale. We prove some new large deviation bounds on this submartingale w~ich enable us to determine the running time of the algorithm.
4,520 citations
"An efficient constraint handling me..." refers background in this paper
...Previous population sizing considerations [21,22] based on schema processing suggested that the population size should increase with the problem size....
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Book•
01 Jan 1973
2,928 citations
Additional excerpts
...Evolutionary strategies (ESs) are evolutionary optimization methods which work on ̄oating-point numbers directly [18,19]....
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Journal Article•
TL;DR: A real-coded crossover operator is developed whose search power is similar to that of the single-point crossover used in binary-coded GAs, and SBX is found to be particularly useful in problems having mult ip le optimal solutions with a narrow global basin where the lower and upper bo unds of the global optimum are not known a priori.
Abstract: Abst ract . T he success of binary-coded gene t ic algorithms (GA s) in problems having discrete sear ch space largely depends on the coding used to represent the prob lem var iables and on the crossover ope ra tor that propagates buildin g blocks from parent strings to children st rings . In solving optimization problems having continuous search space, binary-coded GAs discr et ize the search space by using a coding of the problem var iables in binary strings. However , t he coding of realvalued vari ables in finit e-length st rings causes a number of difficulties: inability to achieve arbit rary pr ecision in the obtained solution , fixed mapping of problem var iab les, inh eren t Hamming cliff problem associated wit h binary coding, and processing of Holland 's schemata in cont inuous search space. Although a number of real-coded GAs are developed to solve optimization problems having a cont inuous search space, the search powers of these crossover operators are not adequate . In t his paper , t he search power of a crossover operator is defined in terms of the probability of creating an arbitrary child solut ion from a given pair of parent solutions . Motivated by the success of binarycoded GAs in discrete search space problems , we develop a real-coded crossover (which we call the simulated binar y crossover , or SBX) operator whose search power is similar to that of the single-point crossover used in binary-coded GAs . Simulation results on a nu mber of realvalued test problems of varying difficulty and dimensionality suggest t hat the real-cod ed GAs with the SBX operator ar e ab le to perfor m as good or bet ter than binary-cod ed GAs wit h the single-po int crossover. SBX is found to be particularly useful in problems having mult ip le optimal solutions with a narrow global basin an d in prob lems where the lower and upper bo unds of the global optimum are not known a priori. Further , a simulation on a two-var iable blocked function shows that the real-coded GA with SBX work s as suggested by Goldberg
2,702 citations
"An efficient constraint handling me..." refers background in this paper
...Some portions of this study have been performed during the author’s visit to the University of Dortmund, Germany, for which the author acknowledges the support from Alexander von Humboldt Foundation....
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...Comments made by Zbigniew Michalewicz on an earlier version of the paper are highly appreciated....
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