Computational Optimization and Applications

, Volume 52, Issue 3, pp 645–666 | Cite as

A polynomial optimization approach to constant rebalanced portfolio selection

Open Access
Article

Abstract

We address the multi-period portfolio optimization problem with the constant rebalancing strategy. This problem is formulated as a polynomial optimization problem (POP) by using a mean-variance criterion. In order to solve the POPs of high degree, we develop a cutting-plane algorithm based on semidefinite programming. Our algorithm can solve problems that can not be handled by any of known polynomial optimization solvers.

Keywords

Multi-period portfolio optimization Polynomial optimization problem Constant rebalancing Semidefinite programming Mean-variance criterion 

References

  1. 1.
    Algoet, P.H., Cover, T.M.: Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Ann. Probab. 16(2), 876–898 (1988) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cover, T.M.: An algorithm for maximizing expected log investment return. IEEE Trans. Inf. Theory 30(2), 369–373 (1984) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Drud, A.S.: CONOPT—a large scale GRG code. ORSA J. Comput. 6(2), 207–216 (1992) CrossRefGoogle Scholar
  5. 5.
    Fleten, S.-E., Høyland, K., Wallace, S.W.: The performance of stochastic dynamic and fixed mix portfolio models. Eur. J. Oper. Res. 140(1), 37–49 (2002) MATHCrossRefGoogle Scholar
  6. 6.
    Gatermann, K., Parrilo, P.A.: Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra 192(1–3), 95–128 (2004) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Henrion, D., Lasserre, J.B., Löfberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Hibiki, N.: Multi-period stochastic optimization models for dynamic asset allocation. J. Bank. Finance 30(2), 365–390 (2006) CrossRefGoogle Scholar
  9. 9.
    Jansson, L., Lasserre, J.B., Riener, C., Theobald, T.: Exploiting symmetries in SDP-relaxations for polynomial optimization. Preprint, Optimization Online (2006) Google Scholar
  10. 10.
    Kim, S., Kojima, M., Waki, H.: Generalized lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems. SIAM J. Optim. 15(3), 697–719 (2005) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kojima, M., Kim, S., Waki, H.: A general framework for convex relaxation of polynomial optimization problems over cones. J. Oper. Res. Soc. Jpn. 46(2), 125–144 (2003) MathSciNetMATHGoogle Scholar
  12. 12.
    Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Program. 103(1), 45–62 (2005) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Konno, H., Yamamoto, R.: A mean-variance-skewness model: algorithm and applications. Int. J. Theor. Appl. Finance 8(4), 409–423 (2005) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Konno, H., Waki, H., Yuuki, A.: Portfolio optimization under lower partial risk measures. Asia-Pac. Financ. Mark. 9(2), 127–140 (2002) MATHCrossRefGoogle Scholar
  15. 15.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lasserre, J.B.: Convergent semidefinite relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Luenberger, D.G.: Investment Science. Oxford University Press, Oxford (1997) Google Scholar
  18. 18.
    Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, 3rd edn. Springer, Berlin (2008) MATHGoogle Scholar
  19. 19.
    Maranas, C.D., Androulakis, I.P., Floudas, C.A., Berger, A.J., Mulvey, J.M.: Solving long-term financial planning problems via global optimization. J. Econ. Dyn. Control 21(8–9), 1405–1425 (1997) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952) Google Scholar
  21. 21.
    Pang, J.-S., Leyffer, S.: On the global minimization of the value-at-risk. Optim. Methods Softw. 19(5), 611–631 (2004) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2), 293–320 (2003) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002) CrossRefGoogle Scholar
  24. 24.
    Sahinidis, N.V., Tawarmalani, M.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Takano, Y., Gotoh, J.: Constant rebalanced portfolio optimization under nonlinear transaction costs. Asia-Pac. Financ. Mark. 18(2), 191–211 (2011) MATHCrossRefGoogle Scholar
  27. 27.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M., Sugimoto, H.: SparsePOP—a sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Softw. 15(2), 15 (2008) MathSciNetGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Management, Graduate School of Decision Science and TechnologyTokyo Institute of TechnologyMeguro-kuJapan
  2. 2.Department of Econometrics and Operations ResearchTilburg UniversityLE TilburgThe Netherlands

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