On analyzing and detecting multiple optima of portfolio optimization

Yue Qi; Zhihao Wang; Su Zhang

doi:10.3934/jimo.2017048

Article Contents

2018, Volume 14, Issue 1: 309-323. Doi: 10.3934/jimo.2017048

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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
* Corresponding author: Su Zhang

Received: July 2016

Revised: November 2016

Early access: June 2017

Published: January 2018

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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Abstract

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.

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

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Figure 1. A feasible region Z of portfolio selection

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

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

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

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

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

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