An Efficient Algorithm for Computing Hypervolume Contributions

The hypervolume indicator serves as a sorting criterion in many recent multi-objective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a population of size n with d objectives, a solution with minimal hypervolume contribution in time (nd2 log n) for d > 2. This improves all previously published algorithms by a factor of n for all d > 3 and by a factor of for d 3. We also analyze hypervolume indicator based optimization algorithms which remove > 1 solutions from a population of size n . We show that there are populations such that the hypervolume contribution of iteratively chosen solutions is much larger than the hypervolume contribution of an optimal set of solutions. Selecting the optimal set of solutions implies calculating conventional hypervolume contributions, which is considered to be computationally too expensive. We present the first hypervolume algorithm which directly calculates the contribution of every set of solutions. This gives an additive term of in the runtime of the calculation instead of a multiplicative factor of . More precisely, for a population of size n with d objectives, our algorithm can calculate a set of solutions with minimal hypervolume contribution in time (nd2 log n n) for d > 2. This improves all previously published algorithms by a factor of nmin{,d2} for d > 3 and by a factor of n for d 3.

[1]  Anne Auger,et al.  Investigating and exploiting the bias of the weighted hypervolume to articulate user preferences , 2009, GECCO.

[2]  Christian Igel,et al.  Efficient covariance matrix update for variable metric evolution strategies , 2009, Machine Learning.

[3]  Lucas Bradstreet,et al.  Maximising Hypervolume for Selection in Multi-objective Evolutionary Algorithms , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[4]  Eckart Zitzler,et al.  Improving hypervolume-based multiobjective evolutionary algorithms by using objective reduction methods , 2007, 2007 IEEE Congress on Evolutionary Computation.

[5]  Nicola Beume,et al.  SMS-EMOA: Multiobjective selection based on dominated hypervolume , 2007, Eur. J. Oper. Res..

[6]  Tobias Friedrich,et al.  Approximating the least hypervolume contributor: NP-hard in general, but fast in practice , 2008, Theor. Comput. Sci..

[7]  Marco Laumanns,et al.  Performance assessment of multiobjective optimizers: an analysis and review , 2003, IEEE Trans. Evol. Comput..

[8]  R. Lyndon While,et al.  A faster algorithm for calculating hypervolume , 2006, IEEE Transactions on Evolutionary Computation.

[9]  Ning Mao,et al.  A Fast Algorithm for Computing the Contribution of a Point to the Hypervolume , 2007, Third International Conference on Natural Computation (ICNC 2007).

[10]  Nicola Beume,et al.  An EMO Algorithm Using the Hypervolume Measure as Selection Criterion , 2005, EMO.

[11]  Joshua D. Knowles.  Local-search and hybrid evolutionary algorithms for Pareto optimization , 2002 .

[12]  Weicheng Xie,et al.  Convergence of multi-objective evolutionary algorithms to a uniformly distributed representation of the Pareto front , 2011, Inf. Sci..

[13]  Joshua D. Knowles,et al.  Bounded archiving using the lebesgue measure , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[14]  Mark H. Overmars,et al.  New upper bounds in Klee's measure problem , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[15]  Eckart Zitzler,et al.  Indicator-Based Selection in Multiobjective Search , 2004, PPSN.

[16]  Alex A. Freitas,et al.  Evolutionary Computation , 2002 .

[17]  Lothar Thiele,et al.  Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach , 1999, IEEE Trans. Evol. Comput..

[18]  Nicola Beume,et al.  S-Metric Calculation by Considering Dominated Hypervolume as Klee's Measure Problem , 2009, Evolutionary Computation.

[19]  Tobias Friedrich,et al.  Don't be greedy when calculating hypervolume contributions , 2009, FOGA '09.

[20]  Nicola Beume,et al.  Multi-objective optimisation using S-metric selection: application to three-dimensional solution spaces , 2005, 2005 IEEE Congress on Evolutionary Computation.

[21]  Marco Laumanns,et al.  Scalable multi-objective optimization test problems , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[22]  Nicola Beume,et al.  On the Complexity of Computing the Hypervolume Indicator , 2009, IEEE Transactions on Evolutionary Computation.

[23]  Tobias Friedrich,et al.  Approximating the volume of unions and intersections of high-dimensional geometric objects , 2008, Comput. Geom..

[24]  Eckart Zitzler,et al.  HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization , 2011, Evolutionary Computation.

[25]  Stefan Roth,et al.  Covariance Matrix Adaptation for Multi-objective Optimization , 2007, Evolutionary Computation.

[26]  Lucas Bradstreet,et al.  Heuristics for optimizing the calculation of hypervolume for multi-objective optimization problems , 2005, 2005 IEEE Congress on Evolutionary Computation.

[27]  Lothar Thiele,et al.  The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration , 2007, EMO.