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Journal Article

Simulated Binary Crossover for Continuous Search Space.

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
Citations
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Journal ArticleDOI
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


Cites background from "Simulated Binary Crossover for Cont..."

  • ...Unless mutation or crossover operators are capable of creating solutions in the basin of another better attractor, the improvement in the convergence toward the true Pareto-optimal front is not possible....

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Book
01 Jan 2001
TL;DR: This text provides an excellent introduction to the use of evolutionary algorithms in multi-objective optimization, allowing use as a graduate course text or for self-study.
Abstract: From the Publisher: Evolutionary algorithms are relatively new, but very powerful techniques used to find solutions to many real-world search and optimization problems. Many of these problems have multiple objectives, which leads to the need to obtain a set of optimal solutions, known as effective solutions. It has been found that using evolutionary algorithms is a highly effective way of finding multiple effective solutions in a single simulation run. · Comprehensive coverage of this growing area of research · Carefully introduces each algorithm with examples and in-depth discussion · Includes many applications to real-world problems, including engineering design and scheduling · Includes discussion of advanced topics and future research · Features exercises and solutions, enabling use as a course text or for self-study · Accessible to those with limited knowledge of classical multi-objective optimization and evolutionary algorithms The integrated presentation of theory, algorithms and examples will benefit those working and researching in the areas of optimization, optimal design and evolutionary computing. This text provides an excellent introduction to the use of evolutionary algorithms in multi-objective optimization, allowing use as a graduate course text or for self-study.

12,134 citations

DOI
01 Jan 2001
TL;DR: An improved version of SPEA, namely SPEA2, is proposed, which incorporates in contrast to its predecessor a fine-grained fitness assignment strategy, a density estimation technique, and an enhanced archive truncation method.
Abstract: The Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele 1999) is a relatively recent technique for finding or approximating the Pareto-optimal set for multiobjective optimization problems. In different studies (Zitzler and Thiele 1999; Zitzler, Deb, and Thiele 2000) SPEA has shown very good performance in comparison to other multiobjective evolutionary algorithms, and therefore it has been a point of reference in various recent investigations, e.g., (Corne, Knowles, and Oates 2000). Furthermore, it has been used in different applications, e.g., (Lahanas, Milickovic, Baltas, and Zamboglou 2001). In this paper, an improved version, namely SPEA2, is proposed, which incorporates in contrast to its predecessor a fine-grained fitness assignment strategy, a density estimation technique, and an enhanced archive truncation method. The comparison of SPEA2 with SPEA and two other modern elitist methods, PESA and NSGA-II, on different test problems yields promising results.

5,062 citations

Book ChapterDOI
18 Sep 2000
TL;DR: Simulation results on five difficult test problems show that the proposed NSGA-II, in most problems, is able to find much better spread of solutions and better convergence near the true Pareto-optimal front compared to PAES and SPEA--two other elitist multi-objective EAs which pay special attention towards creating a diverse Paretimal front.
Abstract: Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) O(MN3) computational complexity (where M is the number of objectives and N is the population size), (ii) non-elitism approach, and (iii) the need for specifying a sharing parameter. In this paper, we suggest a non-dominated sorting based multi-objective evolutionary algorithm (we called it the Non-dominated Sorting GA-II or NSGA-II) which alleviates all the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN2) 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 five difficult test problems show that the proposed NSGA-II, in most problems, is able to find much better spread of solutions and better convergence near the true Pareto-optimal front compared to PAES and SPEA--two other elitist multi-objective EAs which pay special attention towards creating a diverse Pareto-optimal front. Because of NSGA-II's low computational requirements, elitist approach, and parameter-less sharing approach, NSGA-II should find increasing applications in the years to come.

4,878 citations


Cites methods from "Simulated Binary Crossover for Cont..."

  • ...The variables are treated as real numbers and the simulated binary crossover (SBX) [2] and the real-parameter mutation operator are used....

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  • ...The variables are treate d as real numbers and the simulated binary crossover (SBX) [2] and the real-param eter mutation operator are used....

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Journal ArticleDOI
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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01 Jan 1991
TL;DR: Evolution Strategies are algorithms which imitate the principles of natural evolution as a method to solve parameter optimization problems and adaptation of the strategy parameters for the mutation variances as well as their covariances are described.
Abstract: Similar to Genetic Algorithms, Evolution Strategies (ESs) are algorithms which imitate the principles of natural evolution as a method to solve parameter optimization problems. The development of Evolution Strategies from the rst mutation{selection scheme to the reened (,){ES including the general concept of self{adaptation of the strategy parameters for the mutation variances as well as their covariances are described.

876 citations

Journal Article
TL;DR: Results suggest how the sizing equation may be viewed as a coarse delineation of a boundary between what a physicist might call two distinct phases of GA behavior, and how these results may one day lead to rigorous proofs of convergence for recombinative G As operating on problems of bounded description.
Abstract: This paper considers the effect of stochasticity on the quality of convergence of genetic algorithms (GAs). In many problems, the variance of building-block fitness or so-called collateral noise is the major source of variance, and a population-sizing equation is derived to ensure that average signal-to-collateral-noise ratios are favorable to the discrimination of the best building blocks required to solve a problem of bounded deception. The sizing relation is modified to permit the inclusion of other sources of stochasticity, such as the noise of selection, the noise of genetic operators, and the explicit noise or nondeterminism of the objective function. In a test suite of five functions, the sizing relation proves to be a conservative predictor of average correct convergence, as long as all major sources of noise are considered in the sizing calculation. These results suggest how the sizing equation may be viewed as a coarse delineation of a boundary between what a physicist might call two distinct phases of GA behavior. At low population sizes the GA makes many errors of decision, and the quality of convergence is largely left to the vagaries of chance or the serial fixup of flawed results through mutation or other serial injection of diversity. At large population sizes, GAs can reliably discriminate between good and bad building blocks, and parallel processing and recombination of building blocks lead to quick solution of even difficult deceptive problems. Additionally, the paper outlines a number of extensions to this work, including the development of more refined models of the relation between generational average error and ultimate convergence quality, the development of online methods for sizing populations via the estimation of populations via the estimation of population-sizing parameters, and the investigation of populationsizing in the context of niching and other schemes designed for use in problems with high cardinality solution sets. The paper also discusses how these results may one day lead to rigorous proofs of convergence for recombinative G As operating on problems of bounded description.

697 citations


"Simulated Binary Crossover for Cont..." refers background in this paper

  • ..., [6]), proper population sizing is import an t for the successful work ing of GAs ....

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  • ...In bin ary-coded GAs, a larger popu lation is required to achieve a desired solution accuracy for a problem coded in a bigger st ring [6]....

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01 Jan 1991
TL;DR: Intrinsic parallelism is shown to have application beyond schemata and o-schemata and more general objects called formae are introduced and general operators which manipulate these are introduced.
Abstract: Intrinsic parallelism is shown to have application beyond schemata and o-schemata. More general objects called formae are introduced and general operators which manipulate these are introduced and discussed. These include random, respectful recombination. The extended formalism is applied to various common representations and standard operators are analysed in the light of the formalism.

258 citations


"Simulated Binary Crossover for Cont..." refers background or methods or result in this paper

  • ...lThe design of a crossover ope rator depends on a numb er of ot her factors such as respect, assortment , and tra nsmissibility of formae (simil arity temp lates) as suggested in [7] ....

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  • ...It is worth mentioning here that in [7] an ergodicity crit erion is also defined as the ab ility to access any sear ch point through genetic operators from any population....

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  • ...It may be mentioned here that t he interval schemata are conceptually similar to the vir tual alphabets introduced in [10] and locality form ae defined in [7] ....

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  • ...A number of crit eria for the successful design of a crossover operator ar e suggested in [7] ....

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  • ...Since such schemata would be a vari ati on of the int erval schemata or locality form ae, they would also sat isfy the design criteria in [7]....

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