T. Hanne. On the convergence of multiobjective evolutionary algorithms. European J. Of Operational Research, 117:553--564, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ad dressed for example by van Veldhuizen [17] Very re cently an interesting approach has been suggested by Thiele et al. 16] to define a simple problem class for ....

T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Op- erational Research, 117(3):553-564, 1999.

....to a fixed ma.ximum number of individuals, presumably to avoid the computational costs of maintaining a large archive. It is shown in this paper that a consequence of restricting the number of solutions in the elite front can be shrinking [19] and oscillating retreating estimated Pareto fronts [20, 21, 22]. These problems also occur in SPEA, PAES and other existing MOEAs which use a truncated elite archive. A remedy to this situation is simply to retain all the non dominated solutions found (as an active input to the continuing search process) as, for example, used by Parks and Miller [3] ....

....behind (is dominated by) elements of the original archive set (Figure la) Repetitions of this process can lead to the estimated front retreating or, more commonly, oscillating as the front advances in the MOEA search stage but retreats during truncation. This possibility was first noted by Hanne [20] in different situation. In Hanne s MOEA (an applied example of which is in [23] a ES( scheme was used, with a child replacing the parent if the parent does not dominate the child. As such the population that Hanne used was not a Pareto archive as it was not a non dominated set (eq. 7) ....

T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, (117):553-564, 1999.

....design the relatively simple objective function being fixed. A single pbest is maintained for each swarm member, which is only replaced when a new solution is found which is lower on all objectives (identical to the conservative preservation of efficiency selection rule described by I Ianne in [5]) Their model was used on a number of test functions from the lit erature, however no comparison was made with any other models, or the true Pareto fronts for the problems. Parsopoulos and Vrahatis [11] introduce two methods that use a weighted aggregate approach and another that is loosely ....

....as this data structure facilitates the rapid selection of an appropriate archive member for this new multi objective PSO method. 4 Dominated trees Recent studies have highlighted the theoretic inefficiency caused by representing a non dominated set with a limited number of solu tions [5, 9]. This in turn led Fieldsend et al. 4] and Everson et al. 3] to empirically demon strate the inefficiency caused by truncation of estimated Pareto archives in MOEAs, and develop a number of data structures to facilitate the maintenance of unconstrained archives. In this section we shall briefly ....

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T. Hanne. On the convergence of multiob- jective evolutionary algorithms. European Journal of Operational Research, 117:553564, 1999.

....test function design the relatively simple objective function being fixed. A single pbest is maintained for each swarm member, which is only replaced when a new solution is found which dominates it (identical to the conservative preservation of efficiency selection rule described by Hanne in [11]) The performance of the MOPSO was demonstrated on a number of test functions from the literature (including the ZDT test functions from [29] however no comparison was made with any other models, or the true Pareto fronts for the problems. 3.2.2 Parsopoulos and Vrahatis Parsopoulos and ....

T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, 117:553-564, 1999. 24

.... 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 ....

T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, 117(3):553-564, September 2000.

....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. ....

Thomas Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal Of Operational Research, 117(3):553--564, 1999.

.... [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 ....

T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, 117(3):553--564, 1999.

....this algorithm as an example to demonstrate theoretic results about convergence to the Pareto set. It is interesting to note that the algorithm for which the derived properties are valid is an elitist EA. Additional theory about different properties of selection schemes in MOEAs can be found in (Hanne 1999). Certainly many more versions of elitist MOEAs exist 1 , but these recent examples already show the great variety in 1 See, e.g. Horn 1997; Veldhuizen 1999) the implementation of elitism. In the next section we construct a model to provide a general framework for elitism in multi objective ....

Hanne, T. (1999, Sep.). On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research 117(3), 553--564.

....objective functions to combine them into a scalar fitness function. The weight values are generated randomly for each iteration ensuring a good distribution of solutions along the nondominated frontier. Others papers concerning GA and EA (evolutionary algorithms) based procedures are discussed in [16, 37, 35, 156, 122, 121, 154]. 5.2.4 Other Approaches and New Developments Other adaptations of heuristic procedures are found like dedicated heuristics [168] a stochastic search method [306] neural network based methods [199] 303] or the GRASP 19 method [102] We mention also a paper concerning a comparison of ....

T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, 117:553--564, 1999.

....given. 1 Introduction Theoretical results on multi objectiveevolutionary algorithms are scarce. This work extends the results given in Rudolph (1998a) and van Veldhuizen (1999) for finitely large search spaces. Related work treating continuous search spaces may be found in Rudolph (1998b) and Hanne (1999). The plan is as follows: It is assumed that the evolutionary algorithms are Markov processes which have to cope with partially ordered fitness values this includes optimization under a single objective function as well as multiple objective functions. Therefore section 2 recalls some ....

T. Hanne (1999). On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research 117(3), 553--564.

....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) ....

T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, 1999 (to appear).

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T. Hanne. On the convergence of multiobjective evolutionary algorithms. European J. Of Operational Research, 117:553--564, 1999.

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T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, 117:553--564, 1999.

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T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal Of Operational Research, 117(3):553-564, 1999. 26

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Thomas Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research, 117(3):553-564, 1999.

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T. Hanne. On th e convergence of multiobjective evolutionary algorith ms. European Journal Of Operational Research, 117(3):553--564, 1 .

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