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On analyzing and detecting multiple optima of portfolio optimization

  • * Corresponding author: Su Zhang

    * Corresponding author: Su Zhang
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.
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  • 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.

    Mathematics Subject Classification: Primary: 90B50; Secondary: 90C29.


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  • Figure 1.  A feasible region Z of portfolio selection

    Figure 2.  The S, E, Z, and N of the example in subsection 3.1

    Figure 4.  The Z and N of the example in subsection 3.2

    Figure 3.  Incorrectly approximating the N of the example in subsection 3.1 by the major style of portfolio optimization

    Figure 5.  Incorrectly approximating the N of the example in subsection 3.2 by the major style of portfolio optimization

    Figure 6.  The Z and N of the generalization of the example in subsection 3.2

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