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January 2018, 14(1): 309-323. doi: 10.3934/jimo.2017048

## On analyzing and detecting multiple optima of portfolio optimization

 1 China Academy of Corporate Governance & Department of Financial Management, Business School & Collaborative Innovation Center for China Economy, Nankai University, 94 Weijin Road, Tianjin, 300071, China 2 Department of Financial Management, Business School, Nankai University, 94 Weijin Road, Tianjin, 300071, China

* Corresponding author: Su Zhang

Received  July 2016 Revised  November 2016 Published  June 2017

Fund Project: The first author is supported by Social Science Grant of the Ministry of Education of China grant 14JJD630007. The third author is supported by National Natural Science Foundation of China grant 11401322 and Fundamental Research Funds for the Central Universities grant NKZXB1447

Portfolio selection is widely recognized as the birth-place of modern finance; portfolio optimization has become a developed tool for portfolio selection by the endeavor of generations of scholars. Multiple optima are an important aspect of optimization. Unfortunately, there is little research for multiple optima of portfolio optimization. We present examples for the multiple optima, emphasize the risk of overlooking the multiple optima by (ordinary) quadratic programming, and report the software failure by parametric quadratic programming. Moreover, we study multiple optima of multiple-objective portfolio selection and prove the nonexistence of the multiple optima of an extension of the model of Merton. This paper can be a step-stone of studying the multiple optima.

Citation: Yue Qi, Zhihao Wang, Su Zhang. On analyzing and detecting multiple optima of portfolio optimization. Journal of Industrial & Management Optimization, 2018, 14 (1) : 309-323. doi: 10.3934/jimo.2017048
##### References:
 [1] Z. Bodie, A. Kane and A. J. Marcus, Investments, 10th edition, McGraw-Hill/Irwin, Boston, 2013. [2] S. P. Boyd and L. Vandenberghe, Convex Optimization, 1st edition, Cambridge University Press, Cambridge, Britain, 2004. doi: 10.1017/CBO9780511804441. [3] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, 1st edition, Springer Verlag, New York, 1987. doi: 10.1007/978-1-4899-0004-3. [4] V. DeMiguel, L. Garlappi, F.J. Nogales and R. Uppal, A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms, Management Science, 55 (2009), 798-812. [5] V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953. doi: 10.1093/acprof:oso/9780199744282.003.0034. [6] D. Disatnik and S. Katz, Portfolio optimization using a block structure for the covariance matrix, Journal of Business Finance & Accounting, 39 (2012), 806-843. [7] P. H. Dybvig, Short sales restrictions and kinks on the mean variance frontier, The Journal of Finance, 39 (1984), 239-244. doi: 10.1111/j.1540-6261.1984.tb03871.x. [8] M. Ehrgott, Multicriteria Optimization, Volume 491 of Lecture Notes in Economics and Mathematical Systems, 2nd edition, Springer Verlag, Berlin, 2005. doi: 10.1007/978-3-662-22199-0. [9] E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9th edition, John Wiley & Sons, New York, 2014. [10] J. B. Guerard, Handbook of Portfolio Construction: Contemporary Applications of Markowitz Techniques, Springer, New York, 2010. [11] G. Hadley, Nonlinear and Dynamic Programming, Addison Wesley, Reading, Massachusetts, 1964. [12] F. S. Hillier and G. J. Lieberman, Introduction to Operations Research, Third edition. Holden-Day, Inc., Oakland, Calif., 1980. [13] M. Hirschberger, Y. Qi and R. E. Steuer, Large-scale MV efficient frontier computation via a procedure of parametric quadratic programming, European Journal of Operational Research, 204 (2010), 581-588. doi: 10.1016/j.ejor.2009.11.016. [14] C. Huang and R. H. Litzenberger, Foundations for Financial Economics, Prentice Hall, Englewood Cliffs, New Jersey, 1988. [15] B. I. Jacobs, K. N. Levy and H. M. Markowitz, Portfolio optimization with factors, scenarios, and realistic short positions, Operations Research, 53 (2005), 586-599. doi: 10.1287/opre.1050.0212. [16] H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. doi: 10.1111/j.1540-6261.1952.tb01525.x. [17] H. M. Markowitz and A. F. Perold, Portfolio analysis with factors and scenarios, The Journal of Finance, 36 (1981), 871-877. doi: 10.1111/j.1540-6261.1981.tb04889.x. [18] H. M. Markowitz and G. P. Todd, Mean-Variance Analysis in Portfolio Choice and Capital Markets, Frank J. Fabozzi Associates, New Hope, Pennsylvania, 2000. [19] R. C. Merton, An analytical derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872. [20] K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, 1999. [21] Y. Qi, R. E. Steuer and M. Wimmer, An analytical derivation of the efficient surface in portfolio selection with three criteria, Annals of Operations Research, 251 (2017), 161-177. doi: 10.1007/s10479-015-1900-y. [22] M. Rubinstein, Markowitz's "portfolio selection": A fifty-year retrospective, The Journal of Finance, 57 (2002), 1041-1045. [23] M. Stein, J. Branke and H. Schmeck, Efficient implementation of an active set algorithm for large-scale portfolio selection, Computers & Operations Research, 35 (2008), 3945-3961. doi: 10.1016/j.cor.2007.05.004. [24] R. E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley & Sons, New York, 1986. [25] R. E. Steuer and P. Na, Multiple criteria decision making combined with finance: A categorized bibliography, European Journal of Operational Research, 150 (2003), 496-515. doi: 10.1016/S0377-2217(02)00774-9. [26] J. Vörös, The explicit derivation of the efficient portfolio frontier in the case of degeneracy and general singularity, European Journal of Operational Research, 32 (1987), 302-310. doi: 10.1016/S0377-2217(87)80153-4. [27] J. Vörös, J. Kriens and L. W. G. Strijbosch, A note on the kinks at the mean variance frontier, European Journal of Operational Research, 112 (1999), 236-239. [28] P. Wolfe, The simplex method for quadratic programming, Econometrica, 27 (1959), 382-398. doi: 10.2307/1909468. [29] C. Zopounidis, E. Galariotis, M. Doumpos, S. Sarri and K. Andriosopoulos, Multiple criteria decision aiding for finance: An updated bibliographic survey, European Journal of Operational Research, 247 (2015), 339-348. doi: 10.1016/j.ejor.2015.05.032.

show all references

##### References:
 [1] Z. Bodie, A. Kane and A. J. Marcus, Investments, 10th edition, McGraw-Hill/Irwin, Boston, 2013. [2] S. P. Boyd and L. Vandenberghe, Convex Optimization, 1st edition, Cambridge University Press, Cambridge, Britain, 2004. doi: 10.1017/CBO9780511804441. [3] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, 1st edition, Springer Verlag, New York, 1987. doi: 10.1007/978-1-4899-0004-3. [4] V. DeMiguel, L. Garlappi, F.J. Nogales and R. Uppal, A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms, Management Science, 55 (2009), 798-812. [5] V. DeMiguel, L. Garlappi and R. Uppal, Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?, Review of Financial Studies, 22 (2009), 1915-1953. doi: 10.1093/acprof:oso/9780199744282.003.0034. [6] D. Disatnik and S. Katz, Portfolio optimization using a block structure for the covariance matrix, Journal of Business Finance & Accounting, 39 (2012), 806-843. [7] P. H. Dybvig, Short sales restrictions and kinks on the mean variance frontier, The Journal of Finance, 39 (1984), 239-244. doi: 10.1111/j.1540-6261.1984.tb03871.x. [8] M. Ehrgott, Multicriteria Optimization, Volume 491 of Lecture Notes in Economics and Mathematical Systems, 2nd edition, Springer Verlag, Berlin, 2005. doi: 10.1007/978-3-662-22199-0. [9] E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9th edition, John Wiley & Sons, New York, 2014. [10] J. B. Guerard, Handbook of Portfolio Construction: Contemporary Applications of Markowitz Techniques, Springer, New York, 2010. [11] G. Hadley, Nonlinear and Dynamic Programming, Addison Wesley, Reading, Massachusetts, 1964. [12] F. S. Hillier and G. J. Lieberman, Introduction to Operations Research, Third edition. Holden-Day, Inc., Oakland, Calif., 1980. [13] M. Hirschberger, Y. Qi and R. E. Steuer, Large-scale MV efficient frontier computation via a procedure of parametric quadratic programming, European Journal of Operational Research, 204 (2010), 581-588. doi: 10.1016/j.ejor.2009.11.016. [14] C. Huang and R. H. Litzenberger, Foundations for Financial Economics, Prentice Hall, Englewood Cliffs, New Jersey, 1988. [15] B. I. Jacobs, K. N. Levy and H. M. Markowitz, Portfolio optimization with factors, scenarios, and realistic short positions, Operations Research, 53 (2005), 586-599. doi: 10.1287/opre.1050.0212. [16] H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. doi: 10.1111/j.1540-6261.1952.tb01525.x. [17] H. M. Markowitz and A. F. Perold, Portfolio analysis with factors and scenarios, The Journal of Finance, 36 (1981), 871-877. doi: 10.1111/j.1540-6261.1981.tb04889.x. [18] H. M. Markowitz and G. P. Todd, Mean-Variance Analysis in Portfolio Choice and Capital Markets, Frank J. Fabozzi Associates, New Hope, Pennsylvania, 2000. [19] R. C. Merton, An analytical derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872. [20] K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, 1999. [21] Y. Qi, R. E. Steuer and M. Wimmer, An analytical derivation of the efficient surface in portfolio selection with three criteria, Annals of Operations Research, 251 (2017), 161-177. doi: 10.1007/s10479-015-1900-y. [22] M. Rubinstein, Markowitz's "portfolio selection": A fifty-year retrospective, The Journal of Finance, 57 (2002), 1041-1045. [23] M. Stein, J. Branke and H. Schmeck, Efficient implementation of an active set algorithm for large-scale portfolio selection, Computers & Operations Research, 35 (2008), 3945-3961. doi: 10.1016/j.cor.2007.05.004. [24] R. E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley & Sons, New York, 1986. [25] R. E. Steuer and P. Na, Multiple criteria decision making combined with finance: A categorized bibliography, European Journal of Operational Research, 150 (2003), 496-515. doi: 10.1016/S0377-2217(02)00774-9. [26] J. Vörös, The explicit derivation of the efficient portfolio frontier in the case of degeneracy and general singularity, European Journal of Operational Research, 32 (1987), 302-310. doi: 10.1016/S0377-2217(87)80153-4. [27] J. Vörös, J. Kriens and L. W. G. Strijbosch, A note on the kinks at the mean variance frontier, European Journal of Operational Research, 112 (1999), 236-239. [28] P. Wolfe, The simplex method for quadratic programming, Econometrica, 27 (1959), 382-398. doi: 10.2307/1909468. [29] C. Zopounidis, E. Galariotis, M. Doumpos, S. Sarri and K. Andriosopoulos, Multiple criteria decision aiding for finance: An updated bibliographic survey, European Journal of Operational Research, 247 (2015), 339-348. doi: 10.1016/j.ejor.2015.05.032.
A feasible region Z of portfolio selection
The S, E, Z, and N of the example in subsection 3.1
The Z and N of the example in subsection 3.2
Incorrectly approximating the N of the example in subsection 3.1 by the major style of portfolio optimization
Incorrectly approximating the N of the example in subsection 3.2 by the major style of portfolio optimization
The Z and N of the generalization of the example in subsection 3.2
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