Karl Bringmann
Tobias Friedrich
ABSTRACT
In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the dominated portion of the objective space.
Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator is indeed equivalent to the overall objective of finding a good approximation of the Pareto front. To address this question, we compare the optimal approximation factor with the approximation factor achieved by sets maximizing the hypervolume indicator. We bound the optimal approximation factor of n points by 1+Θ(1/n) for arbitrary Pareto fronts. Furthermore, we prove that the same asymptotic approximation ratio is achieved by sets of n points that maximize the hypervolume indicator. This shows that the speed of convergence of the approximation ratio achieved by maximizing the hypervolume indicator is asymptotically optimal.
This implies that for large values of n, sets maximizing the hypervolume indicator quickly approach the optimal approximation ratio. Moreover, our bounds show that also for relatively small values of n, sets maximizing the hypervolume indicator achieve a near-optimal approximation ratio.
- A. Auger, J. Bader, D. Brockhoff, and E. Zitzler. Investigating and exploiting the bias of the weighted hypervolume to articulate user preferences. In Proc. 11th annual Conference on Genetic and Evolutionary Computation (GECCO '09), pp. 563--570, 2009. Google Scholar
Digital Library
- A. Auger, J. Bader, D. Brockhoff, and E. Zitzler. Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point. In Proc. 10th International Workshop on Foundations of Genetic Algorithms (FOGA '09), pp. 87--102, 2009. Google ScholarDigital Library
- J. Bader and E. Zitzler. HypE: An algorithm for fast hypervolume-based many-objective optimization. Evolutionary Computation, 2010+. To appear. Google ScholarDigital Library
- N. Beume. S-Metric calculation by considering dominated hypervolume as Klee's measure problem. Evolutionary Computation, 17: 477--492, 2009. Google ScholarDigital Library
- N. Beume and G. Rudolph. Faster S-metric calculation by considering dominated hypervolume as Klee's measure problem. In Proc. Second International Conference on Computational Intelligence (IASTED '06), pp. 233--238, 2006.Google Scholar
- N. Beume, B. Naujoks, and M. T. M. Emmerich. SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research, 181: 1653--1669, 2007.Google ScholarCross Ref
- K. Bringmann and T. Friedrich. Approximating the volume of unions and intersections of high-dimensional geometric objects. In Proc. 19th International Symposium on Algorithms and Computation (ISAAC '08), Vol. 5369 of LNCS, pp. 436--447, 2008. Google ScholarDigital Library
- K. Bringmann and T. Friedrich. Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. In Proc. 5th International Conference on Evolutionary Multi-Criterion Optimization (EMO '09), pp. 6--20, 2009. Google ScholarDigital Library
- T. C. E. Cheng, A. Janiak, and M. Y. Kovalyov. Bicriterion single machine scheduling with resource dependent processing times. SIAM J. on Optimization, 8: 617--630, 1998. Google ScholarDigital Library
- C. Daskalakis, I. Diakonikolas, and M. Yannakakis. How good is the Chord algorithm? In Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '10), pp. 978--991, 2010. Google ScholarDigital Library
- K. Deb, M. Mohan, and S. Mishra. Evaluating the ε-domination based multi-objective evolutionary algorithm for a quick computation of pareto-optimal solutions. Evolutionary Computation, 13: 501--525, 2005. Google ScholarDigital Library
- I. Diakonikolas and M. Yannakakis. Small approximate pareto sets for biobjective shortest paths and other problems. SIAM Journal on Computing, 39: 1340--1371, 2009.Google ScholarDigital Library
- M. T. M. Emmerich, N. Beume, and B. Naujoks. An EMO algorithm using the hypervolume measure as selection criterion. In Proc. Third International Conference on Evolutionary Multi-Criterion Optimization (EMO '05), pp. 62--76, 2005. Google ScholarDigital Library
- T. Friedrich, C. Horoba, and F. Neumann. Multiplicative approximations and the hypervolume indicator. In Proc. 11th annual Conference on Genetic and Evolutionary Computation (GECCO '09), pp. 571--578, 2009. Google ScholarDigital Library
- C. Igel, N. Hansen, and S. Roth. Covariance matrix adaptation for multi-objective optimization. Evolutionary Computation, 15: 1--28, 2007. Google ScholarDigital Library
- J. D. Knowles and D. Corne. Properties of an adaptive archiving algorithm for storing nondominated vectors. IEEE Trans. Evolutionary Computation, 7: 100--116, 2003. Google ScholarDigital Library
- J. D. Knowles, D. W. Corne, and M. Fleischer. Bounded archiving using the Lebesgue measure. In Proc. IEEE Congress on Evolutionary Computation (CEC '03), Vol. 4, pp. 2490--2497, 2003.Google ScholarCross Ref
- R. Kumar and N. Banerjee. Running time analysis of a multiobjective evolutionary algorithm on simple and hard problems. In Proc. 8th International Workshop on Foundations of Genetic Algorithms (FOGA '05), Vol. 3469 of Lecture Notes in Computer Science, pp. 112--131. Springer, 2005. Google ScholarDigital Library
- M. Laumanns, L. Thiele, K. Deb, and E. Zitzler. Combining convergence and diversity in evolutionary multiobjective optimization. Evolutionary Computation, 10(3): 263--282, 2002. Google ScholarDigital Library
- G. Lizarraga-Lizarraga, A. Hernandez-Aguirre, and S. Botello-Rionda. G-metric: an m-ary quality indicator for the evaluation of non-dominated sets. In Proc. 10th annual Conference on Genetic and Evolutionary Computation (GECCO '08), pp. 665--672, 2008. Google ScholarDigital Library
- B. Naujoks, N. Beume, and M. T. M. Emmerich. Multi-objective optimisation using S-metric selection: application to three-dimensional solution spaces. In Proc. IEEE Congress on Evolutionary Computation (CEC '05), pp. 1282--1289, 2005.Google ScholarCross Ref
- C. H. Papadimitriou and M. Yannakakis. On the approximability of trade-offs and optimal access of web sources. In Proc. 41st annual Symposium on Foundations of Computer Science (FOCS '00), pp. 86--92, 2000. Google ScholarDigital Library
- C. H. Papadimitriou and M. Yannakakis. Multiobjective query optimization. In Proc. 20th ACM Symposium on Principles of Database Systems (PODS '01), pp. 52--59, 2001. Google ScholarDigital Library
- W. Rudin. Real and complex analysis. McGraw-Hill Book Co., New York, 3rd edition, 1987. Google ScholarDigital Library
- S. Vassilvitskii and M. Yannakakis. Efficiently computing succinct trade-off curves. Theor. Comput. Sci., 348: 334--356, 2005. Google ScholarDigital Library
- R. L. While, L. Bradstreet, L. Barone, and P. Hingston. Heuristics for optimizing the calculation of hypervolume for multi-objective optimization problems. In Proc. IEEE Congress on Evolutionary Computation (CEC '05), pp. 2225--2232, 2005.Google Scholar
- X. Zhou, N. Mao, W. Li, and C. Sun. A fast algorithm for computing the contribution of a point to the hypervolume. In Proc. Third International Conference on Natural Computation (ICNC '07), Vol. 4, pp. 415--420, 2007. Google ScholarDigital Library
- E. Zitzler. Hypervolume metric calculation, 2001. Computer Engineering and Networks Laboratory (TIK), ETH Zurich, Switzerland, See ftp:/!/ftp.tik.ee.ethz.ch/pub/people/zitzler/hypervol.c.Google Scholar
- E. Zitzler and S. Künzli. Indicator-based selection in multiobjective search. In Proc. 8th International Conference Parallel Problem Solving from Nature (PPSN VIII), Vol. 3242 of LNCS, pp. 832--842, 2004.Google ScholarCross Ref
- E. Zitzler and L. Thiele. Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans. Evolutionary Computation, 3: 257--271, 1999. Google ScholarDigital Library
- E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. Grunert da Fonseca. Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evolutionary Computation, 7: 117--132, 2003. Google ScholarDigital Library
- E. Zitzler, D. Brockhoff, and L. Thiele. The hypervolume indicator revisited: On the design of Pareto-compliant indicators via weighted integration. In Proc. Fourth International Conference on Evolutionary Multi-Criterion Optimization (EMO '07), Vol. 4403 of LNCS, pp. 862--876, 2007. Google ScholarDigital Library
Index Terms
- The maximum hypervolume set yields near-optimal approximation
Recommendations
Set-based multi-objective optimization, indicators, and deteriorative cycles
GECCO '10: Proceedings of the 12th annual conference on Genetic and evolutionary computationEvolutionary multi-objective optimization deals with the task of computing a minimal set of search points according to a given set of objective functions. The task has been made explicit in a recent paper by Zitzler et al. [13]. We take an order-...
Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point
FOGA '09: Proceedings of the tenth ACM SIGEVO workshop on Foundations of genetic algorithmsThe hypervolume indicator is a set measure used in evolutionary multiobjective optimization to evaluate the performance of search algorithms and to guide the search. Multiobjective evolutionary algorithms using the hypervolume indicator transform ...
Approximation quality of the hypervolume indicator
In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multi-objective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the ...
Comments