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A local relaxation method for the cardinality constrained portfolio optimization problem

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Abstract

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a number of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints present in practice. The proposed local relaxation algorithm explores the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using a suitable heuristic method such as the centroids of initial clusters or be warm started based on the last output. Results, using a basket of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm can lead to significant performance gain over popular branch-and-cut methods. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency.

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Notes

  1. Here we assume columns of R have zero mean.

  2. The center-of-gravity is defined in the usual sense by assigning a weight of one to each asset on the projected space

  3. In the sense of larger distance measure from the cluster centroid assets.

  4. Note a cluster center is simply a point in the projected space, and does not necessarily coincide with any projected asset returns in this space.

References

  1. Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1995)

    MathSciNet  Google Scholar 

  2. Chang, T.-J., Meade, N., Beasley, J.E., Sharaiha, Y.M.: Heuristics for cardinality constrained portfolio optimisation. Comput. Oper. Res. 27(13), 1271–1302 (2000)

    Article  MATH  Google Scholar 

  3. Clausen, J.: Branch and bound algorithms—principles and examples (2003)

  4. Dakin, T.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8(3), 250–255 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  5. Goldfarb, D., Idnani, A.: Dual and primal-dual methods for solving strictly convex quadratic programs. In: Numerical Analysis. Springer, Berlin (1982)

    Google Scholar 

  6. Hansen, P.R., Huang, Z., Shek, H.H.: Realized GARCH: a joint model for returns and realized measures of volatility. J. Appl. Econom. (2011). doi:10.1002/jae.1234. ISSN 1099-1255

    Google Scholar 

  7. Hastie, T., Tibshirani, R., Friedman, J. (eds.): The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Series in Statistics. Springer, Berlin (2003)

    Google Scholar 

  8. Hosrt, R., Pardalos, P.M. (eds.): Handbook of Global Optimization. Kluwer Academic, Norwel (1995)

    Google Scholar 

  9. IBM: IBM ILOG CPLEX optimization studio V12.2 documentation. IBM ILOG (2010)

  10. Konno, H, Yamamoto, R: Integer programming approaches in mean-risk models. Comput. Manag. Sci. 2(4), 339–351 (2005). doi:10.1007/s10287-005-0038-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Konno, H., Yamazaki, H.: Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Manag. Sci. 37(5), 519–531 (1991)

    Article  Google Scholar 

  12. Land, A.H., Doig, A.G.: An automatic method of solving discrete programming problems. Econometrica 28(3), 497–520 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  13. Leyffer, S.: Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18(3), 295–309 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Loraschi, A., Tettamanzi, A., Tomassini, M., Verda, P.: Distributed genetic algorithms with an application to portfolio selection. In: Artificial Neural Nets and Genetic, pp. 384–387. Springer, Berlin (1995)

    Chapter  Google Scholar 

  15. Markowitz, H.M.: Portfolio selection. J. Finance 7, 77–91 (1952)

    Google Scholar 

  16. Markowitz, H.M. (ed.): Mean-Variance Analysis in Portfolio Choice and Capital Markets. Blackwell, Oxford (1987)

    MATH  Google Scholar 

  17. Murray, W., Shanbhag, V.V.: A local relaxation method for nonlinear facility location problems. Multiscale Optim. Methods Appl. 82, 173–204 (2006)

    Article  MathSciNet  Google Scholar 

  18. Murray, W., Shanbhag, U.V.: A local relaxation approach for the siting of electrical substation. Comput. Optim. Appl. 38(3), 299–303 (2007)

    Article  Google Scholar 

  19. Perold, A.F.: Large scale portfolio optimization. Manag. Sci. 30(10), 1143–1160 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shaw, D.X., Liu, S., Kopman, L.: Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optim. Methods Softw. 23(3), 411–420 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shek, H.H.: Modeling high frequency market order dynamics using self-excited point process. Working paper, Stanford University (2010)

  22. Shek, H.H.: Statistical and algorithm aspects of optimal portfolios. PhD thesis (2010)

  23. Yitzhaki, S.: Stochastic dominance, mean variance, and Gini’s mean difference. Am. Econ. Rev. 72(1), 178–185 (1982)

    Google Scholar 

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Acknowledgements

We thank the referees for their careful reading of the manuscript and for prompting us to improve the clarity of the presentation.

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  1. Stanford University, Stanford, CA, 94305, USA

    Walter Murray & Howard Shek

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  1. Walter Murray
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  2. Howard Shek
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Correspondence to Howard Shek.

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Murray, W., Shek, H. A local relaxation method for the cardinality constrained portfolio optimization problem. Comput Optim Appl 53, 681–709 (2012). https://doi.org/10.1007/s10589-012-9471-1

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