# American Institute of Mathematical Sciences

• Previous Article
Is social responsibility for firms competing on quantity evolutionary stable?
• JIMO Home
• This Issue
• Next Article
Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process
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  January 2018 Early access  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:

show all references

##### References:
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
 [1] Yanqin Bai, Chuanhao Guo. Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems. Journal of Industrial & Management Optimization, 2014, 10 (2) : 543-556. doi: 10.3934/jimo.2014.10.543 [2] Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 383-392. doi: 10.3934/dcdss.2016002 [3] Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333 [4] Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089 [5] Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926 [6] Francisco Odair de Paiva. Multiple solutions for a class of quasilinear problems. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 669-680. doi: 10.3934/dcds.2006.15.669 [7] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [8] Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 25-36. doi: 10.3934/jimo.2018138 [9] Cheng-Dar Liou. Optimization analysis of the machine repair problem with multiple vacations and working breakdowns. Journal of Industrial & Management Optimization, 2015, 11 (1) : 83-104. doi: 10.3934/jimo.2015.11.83 [10] Ji Li, Tie Zhou. Numerical optimization algorithms for wavefront phase retrieval from multiple measurements. Inverse Problems & Imaging, 2017, 11 (4) : 721-743. doi: 10.3934/ipi.2017034 [11] Kai Cai, Guangyue Han. An optimization approach to the Langberg-Médard multiple unicast conjecture. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021001 [12] Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1 [13] Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249 [14] Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128 [15] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear coercive Neumann problems. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1957-1974. doi: 10.3934/cpaa.2009.8.1957 [16] Anran Li, Jiabao Su. Multiple nontrivial solutions to a $p$-Kirchhoff equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 91-102. doi: 10.3934/cpaa.2016.15.91 [17] Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129 [18] Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055 [19] Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721 [20] Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145

2020 Impact Factor: 1.801