Published Sep 28, 2018


Google Scholar
Search GoogleScholar

Andrés Vargas



The averaged Hausdorff distance ∆p is an inframetric, recently introduced in evolutionary multiobjective optimization (EMO) as a tool to measure the optimality of finite size approximations to the Pareto front associated to a multiobjective optimization problem (MOP). Tools of this kind are called performance indicators, and their quality depends on the useful criteria they provide to evaluate the suitability of different candidate solutions to a given MOP. We present here a purely theoretical study of the compliance of the ∆p -indicator to the notion of Pareto optimality. Since ∆p is defined in terms of a modified version of other well- known indicators, namely the generational distance GDp , and the inverted generational distance IGDp , specific criteria for the Pareto compliance of each one of them is discussed in detail. In doing so, we review some previously available knowledge on the behavior of these indicators, correcting inaccuracies found in the literature, and establish new and more general results, including detailed proofs and examples of illustrative situations.


averaged Hausdorff distance, generational distance, inverted generational distance, multiobjective optimization, Pareto optimality, performance indicator

[1] Bogoya J M, Vargas A, Cuate O, Schütze O. A ( p, q)-averaged Hausdorff distance for arbitrary measurable sets. Mathematical and Computational Applications, 23(3)51,2018.
doi: 10.3390/mca23030051

[2] Coello C A C, Cortés N C. Solving multiobjective optimization problems using an artificial immune system. Genetic Programming and Evolvable Machines, 6(2):163–190,2005.
doi: 10.1007/s10710-005-6164-x

[3] Hansen M, Jaszkiewicz A. Evaluating the quality of approximations to the non-dominated set. IMM, Department of Mathematical Modelling, Technical University of Denmark,1998.

[4] Heinonen J. Lectures on analysis on metric spaces. Universitext. Springer-Verlag,NewYork,2001.
doi: 10.1007/978-1-4613-0131-8

[5] Hillermeier C. Non linear Multiobjective Optimization. A Generalized Homotopy Approach. Birkhäuser, 2001.
doi: 10.1007/978-3-0348-8280-4

[6] Ishibuchi H, Masuda H,Tanigaki Y,Nojima Y. Modified Distance Calculation in Generational Distance and Inverted Generational Distance. In:Gaspar-Cunha A, Antunes CH, Coello Coello CA (eds.), Evolutionary Multi-Criterion Optimization 8th International Conference EMO 2015, Guimarães, Portugal. Lecture Notes in Computer Science vol.9019, pp.110–125, Springer, 2015.
doi: 10.1007/978-3-319-15892-1_8

[7] ParetoV. Manual of Political Economy: A Critical and Variorum Edition. Edited by Montesano A, Zanni A, Bruni L, Chipman J S, McLureet M. Oxford University Press, 2014.

[8] Rudolph G, Schütze O, Grimme C, Domínguez-Medina C, Trautmann H. Optimal averaged Hausdorff archives for bi-objective problems: theoretical and numerical results. Computational Optimization and Applications, 64:589–618,2016.
doi: 10.1007/s10589-015-9815-8

[9] Schütze O, Esquivel X, Lara A, Coello C A. Using the averaged Hausdorff distance as a performance measure in evolutionary multiobjective optimization. IEEE Transactionson Evolutionary Computation, 16(4):504–522,2012.
doi: 10.1109/TEVC.2011.2161872

[10] Siwel J, Yew-Soon O, Jie Z, Liang F. Consistencies and contradictions of performance metrics in multi objective optimization. IEEE Transactions on Evolutionary Computation, 44(12):2329–2404,2014.
doi: 10.1109/TCYB.2014.2307319

[11] Sun J Q, Xiong F R, Schütze O,Hernández C. Cell Mapping Methods - Algorithmic Approaches and Applications. Nonlinear Systems and Complexity 99, Springer,2019.
doi: 10.1007/978-981-13-0457-6

[12] Vargas A, Bogoya J. A generalization of the averaged Hausdorff distance. Computación y Sistemas, 22(2):331–345,2018.
doi: 10.13053/CyS-22-2-2950

[13] Veldhuizen D. Multi objective evolutionary algorithms: classifications, analyses, and new innovations.1999. Air Force Institute of Technology, page249,1999.

[14] Zitzler E,Thiele L. Multi objective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Transactionson Evolutionary Computation, 3(4):257–271,1999.
doi: 10.1109/4235.797969

[15] Zitzler E,Thiele L, Laumanns M, Fonseca CM, Da Fonseca V G. Performance assessment of multi objective optimizers: Ananalysis and review. IEEE Transactions on Evolutionary Computation, 7(2):117–132, 2003.
doi: 10.1109/TEVC.2003.810758
How to Cite
Vargas, A. (2018). On the Pareto Compliance of the Averaged Hausdorff Distance as a Performance Indicator. Universitas Scientiarum, 23(3), 333–354.
Matemáticas y Estadística / Mathematics and Statistics / Matemática e Estatística