On Sequential Online Archiving of Objective Vectors
In this paper, we examine the problem of maintaining an approximation of the set of nondominated points visited during a multiobjective optimization, a problem commonly known as archiving. Most of the currently available archiving algorithms are reviewed, and what is known about their convergence and approximation properties is summarized. The main scenario considered is the restricted case where the archive must be updated online as points are generated one by one, and at most a fixed number of points are to be stored in the archive at any one time. In this scenario, the \(\vartriangleleft\)-monotonicity of an archiving algorithm is proposed as a weaker, but more practical, property than negative efficiency preservation. This paper shows that hypervolume-based archivers and a recently proposed multi-level grid archiver have this property. On the other hand, the archiving methods used by SPEA2 and NSGA-II do not, and they may \(\vartriangleleft\)-deteriorate with time. The \(\vartriangleleft\)-monotonicity property has meaning on any input sequence of points. We also classify archivers according to limit properties, i.e. convergence and approximation properties of the archiver in the limit of infinite (input) samples from a finite space with strictly positive generation probabilities for all points. This paper establishes a number of research questions, and provides the initial framework and analysis for answering them.
Keywordsapproximation set archive convergence efficiency preserving epsilon-dominance hypervolume online algorithms
Unable to display preview. Download preview PDF.
- 4.Bringmann, K., Friedrich, T.: Don’t be greedy when calculating hypervolume contributions. In: Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pp. 103–112 (2009)Google Scholar
- 5.Bringmann, K., Friedrich, T.: The maximum hypervolume set yields near-optimal approximation. In: Pelikan, M., Branke, J. (eds.) GECCO 2010, pp. 511–518. ACM Press, New York (2010)Google Scholar
- 10.Hansen, M.P.: Metaheuristics for multiple objective combinatorial optimization. Ph.D. thesis, Institute of Mathematical Modelling, Technical University of Denmark (1998)Google Scholar
- 11.Knowles, J.D.: Local-Search and Hybrid Evolutionary Algorithms for Pareto Optimization. Ph.D. thesis, University of Reading, UK (2002)Google Scholar
- 12.Knowles, J.D., Corne, D.: On metrics for comparing non-dominated sets. In: Proceedings of the 2002 Congress on Evolutionary Computation Conference (CEC 2002), pp. 711–716. IEEE Press, Piscataway (2002)Google Scholar
- 15.Laumanns, M.: Stochastic convergence of random search to fixed size Pareto set approximations. Arxiv preprint arXiv:0711.2949 (2007)Google Scholar
- 17.Rudolph, G., Agapie, A.: Convergence properties of some multi-objective evolutionary algorithms. In: Proceedings of the 2000 Congress on Evolutionary Computation (CEC 2000), vol. 2, pp. 1010–1016. IEEE Press, Piscataway (2000)Google Scholar
- 18.Veldhuizen, D.A.V., Lamont, G.B.: Evolutionary computation and convergence to a Pareto front. In: Koza, J.R. (ed.) Late Breaking Papers at the Genetic Programming 1998 Conference, pp. 221–228. Stanford University Bookstore, Stanford University (1998)Google Scholar
- 19.Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Giannakoglou, K., et al. (eds.) Evolutionary Methods for Design, Optimisation and Control, pp. 95–100. CIMNE, Barcelona (2002)Google Scholar