ABSTRACT
Recently, a convergence proof of stochastic search algorithms toward finite size Pareto set approximations of continuous multi-objective optimization problems has been given. The focus was on obtaining a finite approximation that captures the entire solution set in some suitable sense, which was defined by the concept of ε-dominance. Though bounds on the quality of the limit approximation -- which are entirely determined by the archiving strategy and the value of ε -- have been obtained, the strategies do not guarantee to obtain a gap-free Pareto front approximation. Since such approximations are desirable in certain applications, and the related archiving strategies can be advantageous when memetic strategies are included into the search process, we are aiming in this work for such methods. We present two novel strategies that accomplish this task in the probabilistic sense and under mild assumptions on the stochastic search algorithm. In addition to the convergence proofs we give somenumerical results to visualize the behavior of the different archiving strategies.
- http://paradiseo.gforge.inria.fr.Google Scholar
- K. Deb, Samir Agrawal, Amrit Pratap, and T. Meyarivan. A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In Marc Schoenauer et al., editors, Parallel Problem Solving from Nature (PPSN VI) Lecture Notes in Computer Science Vol. 1917, pages 849--858. Springer, 2000. Google ScholarDigital Library
- T. Hanne. On the convergence of multiobjective evolutionary algorithms. European Journal Of Operational Research 117(3):553--564, 1999.Google ScholarCross Ref
- T. Hestermeyer and O. Oberschelp. Selbstoptimierende Fahrzeugregelung Verhaltensbasierte Adaption. In Intelligente mechatronische Systeme volume 122 of HNI-Verlagsschriftenreihe Heinz Nixdorf Institut, 2003.Google Scholar
- C. Hillermeier. Nonlinear Multiobjective Optimization - A Generalized Homotopy Approach Birkhäuser, 2001.Google Scholar
- J. Knowles and D. Corne. Properties of an adaptive archiving algorithm for storing nondominated vectors. IEEE Transactions on Evolutionary Computation 7(2):100--116, 2003.Google ScholarDigital Library
- M. Laumanns, L. Thiele, K. Deb, andE. Zitzler. Combining convergence and diversity in evolutionary multiobjective optimization. Evolutionary Computation 10(3):263--282, 2002. Google ScholarDigital Library
- A. Pottharst, K. Baptist, O. Schütze, J. Böcker, N. Fröhlecke, and M. Dellnitz. Operating point assignment of a linear motor driven vehicle using multiobjective optimization methods. Proceedings of the 11th International Conference EPE-PEMC 2004, Riga, Latvia., 2004.Google Scholar
- G. Rudolph and A. Agapie. On a multi-objective evolutionary algorithm and its convergence to the Pareto set. In Congress on Evolutionary Computation (CEC2000) pages 1010--1016, 2000.Google Scholar
- S. Sayin. Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Mathematical Programming 87:543--560, 2000.Google ScholarCross Ref
- O. Schütze. Set Oriented Methods for Global Optimization PhD thesis, University of Paderborn, 2004. <http://ubdata.uni-paderborn.de/ediss/17/2004/schuetze/>.Google Scholar
- O. Schütze, A. Dell'Aere, and M. Dellnitz. On continuation methods for the numerical treatment of multi-objective optimization problems. In Jürgen Branke, Kalyanmoy Deb, Kaisa Miettinen, and Ralph E. Steuer, editors, Practical Approaches to Multi-Objective Optimization number 04461 in Dagstuhl Seminar Proceedings. Internationales Begegnungs-und Forschungszentrum (IBFI), Schloss Dagstuhl, Germany, 2005. <http://drops.dagstuhl.de/opus/volltexte/2005/349>.Google Scholar
- Oliver Schütze, Marco Laumanns, Carlos A. Coello Coello, Michael Dellnitz, and El-Ghazali Talbi. Convergence of stochastic search algorithms to finite size Pareto set approximations. Research Report 6063, INRIA, 12 2006.Google Scholar
- K. Witting, B. Schulz, A. Pottharst, M. Dellnitz, J. Böcker, and N. Fröhleke. A new approach for online multiobjective optimization of mechatronical systems. Submitted to the International Journal on Software Tools for Technology Transfer STTT, 2005.Google Scholar
- E. Zitzler, M. Laumanns, and L. Thiele. SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization. In K. C. Giannakoglou et al., editors, Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems (EUROGEN 2001) pages 95--100. International Center for Numerical Methods in Engineering (CIMNE), 2002.Google Scholar
Index Terms
- Convergence of stochastic search algorithms to gap-free pareto front approximations
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