# American Institute of Mathematical Sciences

July  2021, 17(4): 1531-1556. doi: 10.3934/jimo.2020033

## Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets

 1 China Academy of Corporate Governance & Department of Financial Management, Business School, 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  May 2019 Revised  August 2019 Published  July 2021 Early access  February 2020

Fund Project: The research is supported by the National Social Science Fund of China 2018 (Grant No. 18BGL063)

Markowitz proposes portfolio selection as a 2-objective model and emphasizes computing (whole) efficient sets and nondominated sets. Computing the sets has long been a topic in multiple-objective optimization. Researchers have gradually recognized other criteria in addition to variance and expected return. To formulate the additional criteria, researchers propose multiple-objective portfolio selection. However, computing the corresponding efficient set and nondominated set is not fully achieved. Moreover, discovering the sets' properties and utilizing the properties remain typically unanswered.

In this paper, we extend Sharpe's and Merton's model by adding a general linear objective and imposing equality constraints. To optimize the model, we analytically derive the minimum-variance surface (defined later), prove it as a nondegenerate paraboloid, and prove the nondominated set as a paraboloidal segment. We also analytically derive the efficient set and prove it as a 2-dimensional translated cone. We then prove that the set subsumes the efficient set of the corresponding traditional model, so the efficient set expands as the general linear objective is added. Furthermore, constraints can be changed or added. We utilize the translated-cone properties and readily compute the changing constraints' effect on the efficient sets by formulae or linear-equation systems.

Citation: Yue Qi, Xiaolin Li, Su Zhang. Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1531-1556. doi: 10.3934/jimo.2020033
##### References:
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Major methods to solve (1) and (3) in the central and right parts, respectively
The minimum-variance surface
An efficient set and efficient sets under changing constraints in $\mathbb{R}^n$
The existence of many nondominated portfolios for $z_1 = 1$ of (4) for the proof of Theorem 4.5
The nondominated set
The existence of zero-covariance portfolio $\mathbf{{z}}^{zcp}$ for (2)
The result of the hypotheses (24)-(27)
 For (24) For (25) For (26) For (27) mean for $\mathbf{{x}}^e$: 0.0052 mean for $\mathbf{{x}}^e$: 0.0022 mean for $\mathbf{{x}}^e$: 0.0052 mean for $\mathbf{{x}}^e$: 0.0022 mean for $\mathbf{{x}}^n$: 0.0054 mean for $\mathbf{{x}}^n$: 0.0020 mean for $\mathbf{{x}}^p$: 0.0055 mean for $\mathbf{{x}}^p$: 0.0019 p-value: 0.9585 p-value: 0.0130 p-value: 0.9268 p-value: 0.0085 accept $H_0$ reject $H_0$ accept $H_0$ reject $H_0$
 For (24) For (25) For (26) For (27) mean for $\mathbf{{x}}^e$: 0.0052 mean for $\mathbf{{x}}^e$: 0.0022 mean for $\mathbf{{x}}^e$: 0.0052 mean for $\mathbf{{x}}^e$: 0.0022 mean for $\mathbf{{x}}^n$: 0.0054 mean for $\mathbf{{x}}^n$: 0.0020 mean for $\mathbf{{x}}^p$: 0.0055 mean for $\mathbf{{x}}^p$: 0.0019 p-value: 0.9585 p-value: 0.0130 p-value: 0.9268 p-value: 0.0085 accept $H_0$ reject $H_0$ accept $H_0$ reject $H_0$
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