G. Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proc. of the 5th IEEE CEC, pages 511--516, Piscataway, NJ, 1998. IEEE Press.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....where eri is the standard deviation. In the standard ES, new individuals are generated in the following way [1] t) t 1) 1) eri(t) eri(t 1) exp( z) exp( zi) 2) 2 THEORY FOR EMOO Results on the convergence of evolutionary multiobjective optimisation have been presented by Rudolph [12, 13] based on the Markov chain approach which has been successfully used for the analysis of single objective evolutionary algorithms, see e.g. 11] among others. The work by Hanne [6] is also mainly concerned with the convergence of evolutionary multiobjective algorithms. Complexity issues have been ....

G. Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511-516, 1998.

....into the VP allocations [37] ## Optimisation methods based on genetic algorithms, evolutionary strategies and evolutionary programming. Numerous suggestions for Multiobjective evolutionary computation algorithms exist [41, 42, 43] but note that theoretical results are still rather limited [44]. III. METHOD In this study we investigate the reported strength of GA in conjunction with CCO to gain further insight into the problem of BAVP, focusing on single objective optimisation (using the sums of the objective functions of the individual VP s) Note that an initial investigation of the ....

G. Rudolph, "On a multiobjective evolutionary algorithm and its convergence to the Pareto Set", ICEC'98, IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, pp. 511516, 4-9 May 1998.

.... SPEA has been used to explore trade o s of software implementations for DSP algorithms [76] and to solve 0 1 knapsack problems [78] 5 Theory The most important theoretical work related to EMOO has concentrated on two main issues: Studies of convergence towards the Pareto optimum set [53, 54, 33, 34, 65]. Ways to compute appropriate sharing factors (or niche sizes) 36, 35, 25] Obviously, a lot of work remains to be done. It would be very interesting to study, for example, the structure of tness landscapes in multiobjective optimization problems [40, 44] Such study could provide some ....

Gunter Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511-516, Piscataway, New Jersey, 1998. IEEE Press.

....The Sphere Model has been subject to intensive theoretical and empirical investigations with evolution strategies, especially in the context of self adaptation. In a multiobjective environment a two variable version of it was used for empirical evaluation of VEGA (Schaffer 1985) while in (Rudolph 1998) it was used for theoretical convergence analysis. Here, a two (SPH 2) and a three (SPH 3) objective instance are considered. Zitzler, Deb, and Thiele s T 6 (Zitzler, Deb, and Thiele 1999) here referred to as ZDT6, is also unimodal and has a non uniformly distributed objective space, both ....

Rudolph, G. (1998). On a multi-objective evolutionary algorithm and its convergence to the pareto set. Technical Report No. CI-17/98, Department of Computer Science/XI, University of Dortmund.

....coded onto chromosomes using identical numbers of bits to represent each parameter. 3 Statistical Comparison of Multiobjective Optimizers Proper comparison of the results of two multiobjective optimizers is a complex issue. Several different solutions have been put forward in recent years [SD94, Rud98, VL98, HJ98, SNT 99, SFF99, ZDT99] However, we use a technique [KC99a] based on a promising method put forward by Fonseca and Fleming [FF96] in which the set of non dominated solutions generated on an optimization run are taken to define a surface, called the attainment surface, that ....

Gunter Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511--516, Piscataway, New Jersey, 1998. IEEE Press.

....al. 1997; Valenzuela Rend on and UrestiCharre, 1997; Fonseca and Fleming, 1998; Parks and Miller, 1998) In recent years, some researchers have investigated particular topics of evolutionary multiobjective search, such as convergence to the Pareto optimal front (Van Veldhuizen and Lamont, 1998a; Rudolph, 1998), niching (Obayashi et al. 1998) and elitism (Parks and Miller, 1998; Obayashi et al. 1998) while others have concentrated on developing new evolutionary techniques (Laumanns et al. 1998; Zitzler and Thiele, 1999) For a thorough discussion of evolutionary algorithms for multiobjective ....

....ways of meaningful statistical interpretation in contrast to the other studies considered here, and furthermore, their methodology seems to be well suited to visualization of the outcomes of several runs. In the context of investigations on convergence to the Pareto optimal front, some authors (Rudolph, 1998; Van Veldhuizen and Lamont, 1998a) have considered the distance of a given set to the Pareto optimal set in the same way as the function M 1 defined below. The distribution was not taken into account, because the focus was not on this Evolutionary Computation Volume 8, Number 2 179 E. Zitzler, K. ....

[Article contains additional citation context not shown here]

Rudolph, G. (1998). On a multi-objective evolutionary algorithm and its convergence to the pareto set. In IEEE International Conference on Evolutionary Computation (ICEC'98), pages 511--516, IEEE Press, Piscataway, New Jersey.

....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. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511--516, Piscataway, New Jersey, 1998. IEEE Press.

.... and Covas 1997; Valenzuela Rend on and Uresti Charre 1997; Fonseca and Fleming 1998; Parks and Miller 1998) In recent years, some researchers have investigated particular topics of evolutionary multiobjective search, such as convergence to the Pareto optimal front (Veldhuizen and Lamont 1998a; Rudolph 1998), niching (Obayashi, Takahashi, and Takeguchi 1998) and elitism (Parks and Miller 1998; Obayashi, Takahashi, and Takeguchi 1998) while others have concentrated on developing new evolutionary techniques (Laumanns, Rudolph, and Schwefel 1998; Zitzler and Thiele 1999) For a thorough discussion of ....

....ways of meaningful statistical interpretation in contrast to the other studies considered here, and furthermore, their methodology seems to be well suited for visualization of the outcomes of several runs. In the context of investigations on convergence to the Pareto optimal front, some authors (Rudolph 1998; Veldhuizen and Lamont 1998a) have considered the distance of a given set to the Pareto optimal set in the same way as the function M 1 de ned below. The distribution was not taken into account, because the focus was not on this matter. However, in comparative studies distance alone is not ....

[Article contains additional citation context not shown here]

Rudolph, G. (1998). On a multi-objective evolutionary algorithm and its convergence to the pareto set. In IEEE International Conference on Evolutionary Computation (ICEC'98), Piscataway, NJ, pp. 511-516. IEEE.

....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. (1998b). On a Multiobjective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the Fifth IEEE Conference on Evolutionary Computation, pages 511--516, IEEE Service Center, Piscataway, New Jersey.

....significance; almost all proposed MOEAs have been tested using this function. It is also an exemplar of relevant MOP concepts. Secondly, as we have noted elsewhere [18] this MOP allows us to determine an analytical expression for PF true (a curve) through substitution. Third, as noted by Rudolph [15], this MOP s P true is given in closed form. At any resolution, the representation of solutions composing P true is thus easily determined without the necessity of exhaustively enumerating the search space. However, its one decision variable implies a large search space should be used when testing ....

Rudolph, Gunter. "On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set." Proceedings of the 1998 IEEE Conference on Evolutionary Computation. 1998.

....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. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511--516, Piscataway, New Jersey, 1998. IEEE Press.

....that perform better than others on certain classes of optimization problems. First theoretical results for the predatorprey approach to find, within one run of an EA, the whole Pareto set of non dominated solutions of a multiple criteria optimization problem have been presented by Rudolph [39]. For keeping pace with the ongoing research, it might be worthwhile to have an eye on the report series of the above mentioned SFB 531 at the University of Dortmund (see http: sfbCI.informatik.unidortmund. de reiheci.html) 6 CONCLUSION Nobody should forget the good old linear and nonlinear ....

G. Rudolph, On a multi-objective evolutionary algorithm and its convergence to the Pareto set, In

....coded onto chromosomes using identical numbers of bits to represent each parameter. 3 Statistical Comparison of Multiobjective Optimizers Proper comparison of the results of two multiobjective optimizers is a complex issue. Several different solutions have been put forward in recent years [SD94, Rud98, VL98, HJ98, SNT 99, SFF99, ZDT99] However, we use a technique [KC99a] based on a promising method put forward by Fonseca and Fleming [FF96] in which the set of non dominated solutions generated on an optimization run are taken to define a surface, called the attainment surface, that ....

Gunter Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511--516, Piscataway, New Jersey, 1998. IEEE Press.

....in maintaining a diverse pareto optimal solutions. The main aim of this paper is to study which difficulties the MOBEA system has by achieving its performance: robustness of the convergence to the true pareto optimal surface, uniform distribution of the population on it (Zitzler and Thiele, 1998; Rudolph, 1998). 2 STRUCTURE OF THE MOBEA In order to make our MOBEA possible to solve challenging MOPs mentioned above, the following algorithms are implemented in this system. We propose a so called C fitness (degree of (in)feasibility) for effectively evolving the population towards the feasible region: ....

Rudolph, G. (1998). On a multiobjective evolutionary algorithm and its convergence to the pareto set.

....provided some important (general) concepts on Pareto ranking, nondominance, and ways to determine sharing factors and mating restriction parameters. ffl Horn [59, 60] and Horn and Nafpliotis [61] have provided important guidelines to choose appropriate values for the sharing factor. ffl Rudolph [62] and Van Veldhuizen and Lamont [63] have provided some theoretical analysis of convergence towards the Pareto set in an attempt to define the limits of GA based search in this domain. Obviously, a lot of work remains to be done regarding theory of EMOO techniques. First, it would be desirable to ....

Gunter Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511--516, Piscataway, New Jersey, 1998. IEEE Press.

....and it is common practice to apply fixed mutation rates in binary coded representation or fixed mutation step sizes in Evolution Strategies. Theoretical considerations, however, emphasize the importance of the mutation strength for the convergence of multi objective evolutionary algorithms (MOEAs) [Rud98, Han99]. When large search spaces are to be explored, adaptive variation operators are mandatory to achieve both a satisfactory rate of progress towards the optimum and a high precision of solutions. Thus, effective control mechanisms, which exist for the single objective case, need to be developed. ....

....mechanisms for the mutation strength are a necessity for many optimization problems. This is equally important in the case of multiple objectives as can be drawn from theoretic results. Stochastic convergence of a simple (1 1) EA to the Pareto set of a two objective problem has been shown in [Rud98]. However, this could only be guaranteed if the mutation step size was chose proportionally to the distance to the Pareto set. Existing approaches to mutation control can be categorized into three groups of increasing complexity: Predefined schedules without feedback, Feedback based ....

[Article contains additional citation context not shown here]

Gunter Rudolph. On a multi-objective evolutionary algorithm and its convergence to the pareto set. In IEEE Int'l Conf. on Evolutionary Computation (ICEC'98), pages 511--516, Piscataway, 1998. IEEE Press.

.... [7] for a survey) There is also a steadily growing theory for EAs facing a (single) stochastically perturbed objective function as can be learned from the overview presented in [1] 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 ....

G. Rudolph. On a multi--objective evolutionary algorithm and its convergence to the Pareto set. In Proceedings of the 1998 IEEE International Conference on Evolutionary Computation, pages 511--516. IEEE Press, Piscataway (NJ), 1998.

....multi objective optimization has remained comparatively rare. Though most algorithms simply apply standard operators from the single objective case, their behavior in the presence of multiple objectives may differ substantially, particularly concerning the adaptation of mutation intensities [1]. When large search spaces are to be explored, adaptive variation operators are mandatory to achieve both, a satisfactory rate of progress towards the optimum and a high precision of solutions. In this study we explore the behavior of a standard self adaptive evolution strategy (SA ES) with ....

Gunter Rudolph. On a multi-objective evolutionary algorithm and its convergence to the pareto set. In IEEE International Conference on Evolutionary Computation (ICEC'98), pages 511--516, Piscataway, NJ, 1998. IEEE.

....Future work should therefore be engaged in examining other evolutionary algorithms with respect to these properties. Since these (sufficient) conditions were only proved for finite search sets a generalization to infinite search sets is desirable. Some work on such search sets is available [9, 10] albeit specialized to multi criteria problems. It would be instructive to generalize these results to the problem of finding minimal elements of arbitrary partially ordered sets. Acknowledgments This work is a result of the Collaborative Research Center Computational Intelligence (SFB 531) ....

G. Rudolph. On a multi--objective evolutionary algorithm and its convergence to the Pareto set. In Proceedings of the 1998 IEEE International Conference on Evolutionary Computation, pages 511--516. IEEE Press, Piscataway (NJ), 1998.

No context found.

G. Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proc. of the 5th IEEE CEC, pages 511--516, Piscataway, NJ, 1998. IEEE Press.

No context found.

G. Rudolph, "On a multiobjective evolutionary algorithm and its convergence to the Pareto Set", ICEC'98, IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, pp. 511516, 4-9 May 1998.

No context found.

G. Rudolph. On a multi-objective evolutionary algorithm and its convergence to the pareto set. In IEEE Int'l Conf. on Evolutionary Computation (ICEC'98), pages 511-516, Piscataway, 1998. IEEE Press.

No context found.

G. Rudolph. On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In 5th IEEE Conference on Evolutionary Computation, pages 511-516, 1998.

No context found.

Rudolph, G. (1998b). On a Multi-Objective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the Fifth IEEE Conference on Evolutionary Computation, pages 511-- 516, IEEE Press, Piscataway, New Jersey.

No context found.

Rudolph, G. (1998a). On a Multiobjective Evolutionary Algorithm and Its Convergence to the Pareto Set. In Proceedings of the 5th IEEE Conference on Evolutionary Computation, pages 511--516. Piscataway, NJ: IEEE Service Centre.

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