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Hypervolume Sharpe-Ratio Indicator: Formalization and First Theoretical Results

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9921)


Set-quality indicators have been used in Evolutionary Multiobjective Optimization Algorithms (EMOAs) to guide the search process. A new class of set-quality indicators, the Sharpe-Ratio Indicator, combining the selection of solutions with fitness assignment has been recently proposed. This class is based on a formulation of fitness assignment as a Portfolio Selection Problem which sees solutions as assets whose returns are random variables, and fitness as the investment in such assets/solutions. An instance of this class based on the Hypervolume Indicator has shown promising results when integrated in an EMOA called POSEA. The aim of this paper is to formalize the class of Sharpe-Ratio Indicators and to demonstrate some of the properties of that particular Sharpe-Ratio Indicator instance concerning monotonicity, sensitivity to scaling and parameter independence.


  • Sharpe Ratio
  • Portfolio selection
  • Evolutionary algorithms
  • Multiobjective optimization

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  1. Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the hypervolume indicator: optimal \(\mu \)-distributions and the choice of the reference point. In: Foundations of Genetic Algorithms (FOGA 2009), pp. 87–102. ACM (2009)

    Google Scholar 

  2. Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. EJOR 181, 1653–1669 (2007)

    CrossRef  MATH  Google Scholar 

  3. Cornuejols, G., Tuntuncu, R.: Optimization Methods in Finance. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  4. Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  5. Knowles, J.D.: Local-search and hybrid evolutionary algorithms for Pareto optimization. Ph.D. thesis, Department of Computer Science, University of Reading (2002)

    Google Scholar 

  6. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  7. Rudolph, G., Schütze, O., Trautmann, H.: On the closest averaged Hausdorff archive for a circularly convex Pareto front. In: Squillero, G., Burelli, P. (eds.) EvoApplications 2016. LNCS, vol. 9598, pp. 42–55. Springer, Heidelberg (2016). doi:10.1007/978-3-319-31153-1_4

    CrossRef  Google Scholar 

  8. Yevseyeva, I., Guerreiro, A.P., Emmerich, M.T.M., Fonseca, C.M.: A portfolio optimization approach to selection in multiobjective evolutionary algorithms. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds.) PPSN 2014. LNCS, vol. 8672, pp. 672–681. Springer, Heidelberg (2014)

    Google Scholar 

  9. Zitzler, E.: Evolutionary algorithms for multiobjective optimization: methods and applications. Ph.D. thesis, ETH Zurich, Switzerland (1999)

    Google Scholar 

  10. Zitzler, E., Knowles, J.D., Thiele, L.: Quality assessment of pareto set approximations. In: Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.) Multiobjective Optimization. LNCS, vol. 5252, pp. 373–404. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  11. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)

    CrossRef  Google Scholar 

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This work was supported by national funds through the Portuguese Foundation for Science and Technology (FCT), by the European Regional Development Fund (FEDER) through COMPETE 2020 – Operational Program for Competitiveness and Internationalization (POCI). A. P. Guerreiro acknowledges FCT for Ph.D. studentship SFHR/BD/77725/2011, co-funded by the European Social Fund and by the State Budget of the Portuguese Ministry of Education and Science in the scope of NSRF–HPOP–Type 4.1–Advanced Training.

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Correspondence to Andreia P. Guerreiro .

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Guerreiro, A.P., Fonseca, C.M. (2016). Hypervolume Sharpe-Ratio Indicator: Formalization and First Theoretical Results. In: Handl, J., Hart, E., Lewis, P., López-Ibáñez, M., Ochoa, G., Paechter, B. (eds) Parallel Problem Solving from Nature – PPSN XIV. PPSN 2016. Lecture Notes in Computer Science(), vol 9921. Springer, Cham.

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