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Particle Swarm Optimization with non-smooth penalty reformulation for a complex portfolio selection problem


  • Marco Corazza

    () (Department of Economics, Ca� Foscari University of Venice; Advanced School of Economics in Venice.)

  • Giovanni Fasano

    () (Department of Management, Ca� Foscari University of Venice; CNR - INSEAN (Italian Ship Model Basin).)

  • Riccardo Gusso

    () (Department of Economics, Ca� Foscari University of Venice.)


In the classical model for portfolio selection the risk is measured by the variance of returns. It is well known that, if returns are not elliptically distributed, this may cause inaccurate investment decisions. To address this issue, several alternative measures of risk have been proposed. In this contribution we focus on a class of measures that uses information contained both in lower and in upper tail of the distribution of the returns. We consider a nonlinear mixed-integer portfolio selection model which takes into account several constraints used in fund management practice. The latter problem is NP-hard in general, and exact algorithms for its minimization, which are both effective and efficient, are still sought at present. Thus, to approximately solve this model we experience the heuristics Particle Swarm Optimization (PSO). Since PSO was originally conceived for unconstrained global optimization problems, we apply it to a novel reformulation of our mixed-integer model, where a standard exact penalty function is introduced.

Suggested Citation

  • Marco Corazza & Giovanni Fasano & Riccardo Gusso, 2011. "Particle Swarm Optimization with non-smooth penalty reformulation for a complex portfolio selection problem," Working Papers 2011_10, Department of Economics, University of Venice "Ca' Foscari".
  • Handle: RePEc:ven:wpaper:2011_10

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    References listed on IDEAS

    1. Fishburn, Peter C, 1977. "Mean-Risk Analysis with Risk Associated with Below-Target Returns," American Economic Review, American Economic Association, vol. 67(2), pages 116-126, March.
    2. Nikos S. Thomaidis & Timotheos Angelidis & Vassilios Vassiliadis & Georgios Dounias, 2009. "Active Portfolio Management With Cardinality Constraints: An Application Of Particle Swarm Optimization," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 5(03), pages 535-555.
    3. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    4. Willard I. Zangwill, 1967. "Non-Linear Programming Via Penalty Functions," Management Science, INFORMS, vol. 13(5), pages 344-358, January.
    5. Fischer, T., 2003. "Risk capital allocation by coherent risk measures based on one-sided moments," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 135-146, February.
    6. Chen, Zhiping & Wang, Yi, 2008. "Two-sided coherent risk measures and their application in realistic portfolio optimization," Journal of Banking & Finance, Elsevier, vol. 32(12), pages 2667-2673, December.
    7. repec:wsi:ijtafx:v:11:y:2008:i:01:n:s0219024908004713 is not listed on IDEAS
    8. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    9. Borges, Bernhard F. J. & Knetsch, Jack L., 1998. "Tests of market outcomes with asymmetric valuations of gains and losses: Smaller gains, fewer trades, and less value," Journal of Economic Behavior & Organization, Elsevier, vol. 33(2), pages 185-193, January.
    10. Martin R. Young, 1998. "A Minimax Portfolio Selection Rule with Linear Programming Solution," Management Science, INFORMS, vol. 44(5), pages 673-683, May.
    11. Enrique Ballestero, 2005. "Mean-Semivariance Efficient Frontier: A Downside Risk Model for Portfolio Selection," Applied Mathematical Finance, Taylor & Francis Journals, vol. 12(1), pages 1-15.
    12. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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    Cited by:

    1. Marco Corazza & Giacomo Di Tollo & Giovanni Fasano & Raffaele Pesenti, 2015. "A novel initialization of PSO for costly portfolio selection problems," Working Papers 4, Department of Management, Università Ca' Foscari Venezia.
    2. Marco Corazza & Giovanni Fasano & Stefania Funari & Riccardo Gusso, 2017. "PSO-based tuning of MURAME parameters for creditworthiness evaluation of Italian SMEs," Working Papers 04, Department of Management, Università Ca' Foscari Venezia.

    More about this item


    Portfolio selection; coherent risk measure; fund management constraints; NP-hard mathematical programming problem; PSO; exact penalty method; SP100 index's assets.;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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