Gunter Rudolph. Evolutionary Search for Minimal Elements in Partially Ordered Finite Sets. In Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345--353. Springer, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The latter has been compared to some of the most popular MOGAs, on a range of problems and test functions, with very positive results. Some theoretical justification for the use of evolutionary algorithms in multiobjective optimization, in the form of convergence proofs, has also been provided [23, 24]. Almost in parallel to the development of MOGAs, there has been a growing research effort in the use of metaheuristics within the field of multiple criteria decision making (MCDM) a branch of operations research. Algorithms based on both tabu search and simulated annealing have been put ....

G. Rudolph. Evolutionary Search for Minimal Elements in Partially Ordered Finite Sets. In V. Porto, N. Saravanan, D. Waagen, and A. Eiben, editors, Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345--353, Berlin, 1998. Springer-Verlag.

....can be seen as a special case of Multiple Criteria Decision Making (Steuer 1986) Corresponding algorithms serve as a tool of decision analysis. In this context it may be desirable to find or to approximate the Pareto optimal solutions, i.e. the minimal elements of a partially ordered set (Rudolph 1998). The ordering relation is the decision maker s preference relation on the set of decision alternatives. There are, of course, some algorithmic tools devoted to related tasks of decision analysis, for instance goal programming (Deb 1998) or methods of a priori or interactive incorporation of ....

....by Knowles and Corne (1999) is a multi objective (1 1) evolution strategy that uses the dominance relation as the selection criterion. Here, elitism is guaranteed by the plus selection, while the archive is used only as a comparison set for incomparable individuals. The (1 1) EA examined by Rudolph (1998) uses a selection criterion that changes randomly. But since only two individuals are compared, the dominated one will never be preferred. In his study, Rudolph uses this algorithm as an example to demonstrate theoretic results about convergence to the Pareto set. It is interesting to note that ....

Rudolph, G. (1998). Evolutionary search for minimal elements in partially ordered sets. In Evolutionary Programming VII -- Proc. Seventh Annual Conf. on Evolutionary Programming (EP-98), San Diego CA. The MIT Press, Cambridge MA.

....with population based methods. Good results have been obtained with such methods (Czyzak and Jaszkiewicz, 1998; Gandibleux et al. 1996; Hansen, 1997, 1998; Serafini, 1994; Ulungu et al. 1995) and, recently, some theoretical work has been done which yields convergence proofs for simple variants (Rudolph, 1998a, 1998b) However, c fl2000 by the Massachusetts Institute of Technology Evolutionary Computation 8(2) 149 172 J. Knowles and D. Corne it is currently unclear how well local search based multiobjective optimizers compare with evolutionary algorithm based approaches. Here, we introduce a novel evolutionary algorithm called ....

Rudolph, G. (1998a). Evolutionary Search for Minimal Elements in Partially Ordered Finite Sets. In Porto, V. W., Saravanan, N., Waagen, D. and Eiben, A. E., editors, Evolutionary Programming VII, Proceedings of the Seventh Annual Conference on Evolutionary Programming, pages 343--353, Springer, Berlin, Germany.

....has been compared to some of the most popular MOGAs, on a range of problems and test functions, with very positive results. Some theoretical justification has also been provided for the use of evolutionary algorithms in multiobjective optimization, in the form of convergence proofs due to Rudolph [23, 24]. Almost in parallel there has been a growing research effort in the use of metaheuristics within the field of Multiple Criteria Decision Making, a branch of Operations Research. Algorithms based on both Tabu Search and Simulated Annealing have been put forward [3, 7, 8, 10, 26, 28] Most of ....

G. Rudolph. Evolutionary Search for Minimal Elements in Partially Ordered Finite Sets. In V. Porto, N. Saravanan, D. Waagen, and A. Eiben, editors, Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345--353, Berlin, 1998. Springer-Verlag.

....interesting to introduce special features (such as elitism, mutation, or other diversity preserving operators) the presence of which may help us to prove convergence of a GA population to the global Pareto optimal front. Attempts to some such proofs exist for single objective GAs (Suzuki, 1993; Rudolph, 1998) and a similar proof may also be attempted for multi objective GAs. Elitism is an useful and popular mechanism used in single objective GAs. Elitism ensures that the best solutions in each generation will not be lost. They are directly carried over from one generation to the next and what is ....

Rudolph, G. (1998). Evolutionary search for minimal elements in partially ordered finite sets. Evolutionary Programming VII. 345--353.

.... of using a selection procedure that is based on the totally ordered set of noisy fitness values we endow the probabilistic fitness set with an appropriate partial order and deploy EAs with those selection methods being explicitly designed for coping with arbitrary partially ordered fitness sets (Rudolph 1998, 2001a; Rudolph and Agapie 2000) Section 2 offers a brief introduction to partially ordered sets in general and in particular to interval orders (Fishburn 1985) which constitute the first step towards the partial order to be used later on. Since the noise is supposed to have bounded support we ....

G. Rudolph (1998). Evolutionary search for minimal elements in partially ordered finite sets. In V. W. Porto, N. Saravanan, D. Waagen, and A. E. Eiben (Eds.), Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pp. 345--353.

....In case of multiple objective functions, however, the theory is still in its infancy: Only few results are known [8, 4] The situation is even worse for other problem classes since theoretical results concerning EAs are unknown apparently. This situation may change by the approach initiated in [6]. Instead of developing an own theory for each problem class, it suffices to develop a theory for EAs that can cope with partially ordered fitness values since many problems may be seen as special cases of the problem of finding the set of minimal (or maximal) elements in a partially ordered set. ....

G. Rudolph. Evolutionary search for minimal elements in partially ordered finite sets. In V. W. Porto, N. Saravanan, D. Waagen, and A. E. Eiben, editors, Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345--353. Springer, Berlin, 1998.

.... of using a selection procedure that is based on the totally ordered set of noisy fitness values we endow the probabilistic fitness set with an appropriate partial order and deploy EAs with those selection methods being explicitly designed for coping with arbitrary partially ordered fitness sets (Rudolph 1998, 2001; Rudolph and Agapie 2000) Section 2 offers a brief introduction to partially ordered sets in general and in particular to interval orders (Fishburn 1985) which constitute the first step towards the partial order to be used later on. Since the noise is supposed to have bounded support we ....

G. Rudolph (1998). Evolutionary search for minimal elements in partially ordered finite sets. In V. W. Porto, N. Saravanan, D. Waagen, and A. E. Eiben (Eds.), Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pp. 345--353.

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G. Rudolph (1998a). Evolutionary search for minimal elements in partially ordered finite sets. In V. W. Porto, N. Saravanan, D. Waagen, and A. E. Eiben (Eds.), Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pp. 345--353.

....by numerous proposals of multicriteria evolutionary algorithms during the last few years this rapid development was, however, not accompanied by a comparable build up of a theoretical foundation. But the first steps towards an elimination of this shortcoming has been made: It was shown in [1] in case of finite search sets that an evolutionary algorithm (EA) with positive variation kernel and elite preservation strategy (these notions are explained later) is capable of generating a sequence of populations such that at least one individual enters the set of minimal elements of the ....

....that the entire population of the evolutionary algorithm consists of minimal elements after a finite number of steps with probability one. ut It should be mentioned that the evolutionary algorithm considered here realizes a stronger version of the elite preservation strategy than introduced in [1]: Unless there is an offspring that dominates a specific parent, this parent will also be a parent of the next iteration. This stronger version is apparently necessary for proving the convergence of the entire population to the set of minimal elements. An example of an evolutionary algorithm that ....

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G. Rudolph. Evolutionary search for minimal elements in partially ordered finite sets. In V. W. Porto, N. Saravanan, D. Waagen, and A. E. Eiben, editors, Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345--353. Springer, Berlin, 1998.

....of evolutionary algorithms that do not necessarily fit in this framework. In this case, the assumptions and proofs must be adapted. For example, it was recently shown that Corollary 1 can be generalized to situations in which the set of fitness values is only partially in lieu of totally ordered [16]. Actually, only the assumptions regarding the selection methods were generalized. 2.2. Time Inhomogeneous Transitions The development of evolutionary algorithms with time inhomogeneous transitions was motivated by the observation that a specific popular evolutionary algorithm fulfilling the ....

G. Rudolph. Evolutionary search for minimal elements in partially ordered finite sets. In V. W. Porto, N. Saravanan, D. Waagen, and A. E. Eiben, editors, Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345--353. Springer, Berlin, 1998.

.... For example, it can be shown that an EA generates at least one stochastic trajectory converging to the Pareto set with probability one if the search space is finite, the support of the fixed mutation distribution covers the search space, and offsprings are accepted if they dominate the parents [4]. Since these assumption are rather strong it may be instructive to investigate what happens if some of these assumptions are weakened. Here, the analysis will focus on a simplified version of a multi objective EA originally presented in [5] Assume there are two objective functions f 0 and f 1 , ....

G. Rudolph. Evolutionary search for minimal elements in partially ordered finite sets. In Proceedings of the 7th Annual Conference on Evolutionary Programming (EP'98). Springer, Berlin and Heidelberg, 1998.

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Gunter Rudolph. Evolutionary Search for Minimal Elements in Partially Ordered Finite Sets. In Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345--353. Springer, 1998.

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G. Rudolph. Evolutionary search for minimal elements in partially ordered nite sets. In V.W. Porto, N. Saravanan, D. Waagen, and A.E. Eiben, editors, Evolutionary Programming VII, Proceedings of the 7th Annual Conference on Evolutionary Programming, pages 345-353, Berlin, 1998. Springer.

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G. Rudolph Convergence Properties of Evolutionary Algorithms.VerlagDr.Kovac, Hamburg, 1 7. . G. Rudolph . Evolutionary search for minimal elements in partially ordered sets. In Evolutionary Programming VII -- Proc. Seventh Annual Conf. on Evolutionary Programming (EP-98), San Diego CA, 1

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Rudolph, G. (1998a). Evolutionary Search for Minimal Elements in Partially Ordered Finite Sets. In Porto, V. W., Saravanan, N., Waagen, D. and Eiben, A. E., editors, Proceedings of the Seventh Annual Conference on Evolutionary Programming, pages 345-353, Springer, Berlin, Germany.

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