# American Institute of Mathematical Sciences

March  2021, 17(2): 695-709. doi: 10.3934/jimo.2019130

## Robust multi-period and multi-objective portfolio selection

 Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia

* Corresponding author: lydiajianglin@gmail.com

Received  January 2019 Revised  June 2019 Published  March 2021 Early access  October 2019

In this paper, a multi-period multi-objective portfolio selection problem with uncertainty is studied. Under the assumption that the uncertainty set is ellipsoidal, the robust counterpart of the proposed problem can be transformed into a standard multi-objective optimization problem. A weighted-sum approach is then introduced to obtain Pareto front of the problem. Numerical examples will be presented to illustrate the proposed method and validate the effectiveness and efficiency of the model developed.

Citation: Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130
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##### References:
The variation of $g(\kappa)$ in terms of $\kappa$ when $\lambda = 0.1$
The variation of $g(\kappa)$ in terms of $\kappa$ when $\lambda=0.5$
The variation of $g(\kappa)$ in terms of $\kappa$ when $\lambda = 0.9$
The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\delta$ when $\lambda = 0.1$
The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\delta$ when $\lambda = 0.5$
The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\delta$ when $\lambda = 0.9$
The variation of $f_{\lambda}(\Delta w_t, \kappa^*)$ in terms of $\varsigma$ when $\lambda = 0.5$
The pareton front with $\delta = 1$
The pareton front with $\delta = 5$
The pareton front with $\delta = 10$
Solution with five period; $\varsigma = 0.001, \delta = 0.01$
 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ 1.6062540e+002 1.6005381e+002 1.5941714e+002 1.5883923e+002 1.5824019e+002 $\Delta w_2$ -4.3718793e+001 -4.3458314e+001 -4.3193335e+001 -4.2953594e+001 -4.2647292e+001 $\Delta w_3$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_4$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_5$ 1.9724480e+003 1.9722713e+003 1.9720766e+003 1.9718941e+003 1.9717006e+003 $\Delta w_6$ -1.3675740e+002 -1.3651295e+002 -1.3620477e+002 -1.3596164e+002 -1.3572250e+002 $\Delta w_7$ -9.7855928e+002 -9.7836228e+002 -9.7827192e+002 -9.7813905e+002 -9.7802742e+002 $\Delta w_8$ -4.6542445e+002 -4.6521487e+002 -4.6500508e+002 -4.6479322e+002 -4.6454631e+002 $\Delta w_9$ 1.5230230e+003 1.5227606e+003 1.5225158e+003 1.5222595e+003 1.5219708e+003 $\Delta w_{10}$ -3.8955914e+001 -3.8854705e+001 -3.8649778e+001 -3.8458643e+001 -3.8279222e+001
 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ 1.6062540e+002 1.6005381e+002 1.5941714e+002 1.5883923e+002 1.5824019e+002 $\Delta w_2$ -4.3718793e+001 -4.3458314e+001 -4.3193335e+001 -4.2953594e+001 -4.2647292e+001 $\Delta w_3$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_4$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_5$ 1.9724480e+003 1.9722713e+003 1.9720766e+003 1.9718941e+003 1.9717006e+003 $\Delta w_6$ -1.3675740e+002 -1.3651295e+002 -1.3620477e+002 -1.3596164e+002 -1.3572250e+002 $\Delta w_7$ -9.7855928e+002 -9.7836228e+002 -9.7827192e+002 -9.7813905e+002 -9.7802742e+002 $\Delta w_8$ -4.6542445e+002 -4.6521487e+002 -4.6500508e+002 -4.6479322e+002 -4.6454631e+002 $\Delta w_9$ 1.5230230e+003 1.5227606e+003 1.5225158e+003 1.5222595e+003 1.5219708e+003 $\Delta w_{10}$ -3.8955914e+001 -3.8854705e+001 -3.8649778e+001 -3.8458643e+001 -3.8279222e+001
Solution with five period; $\varsigma = 0.01, \delta = 0.01$
 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ 2.1589291e-002 -3.4351365e-002 5.7805964e-002 2.3686226e-003 1.0158904e-003 $\Delta w_2$ -2.8283011e-001 -2.7323758e-004 1.6033220e-003 -3.5364786e-003 -2.6401259e-004 $\Delta w_3$ -3.3036266e+002 -3.5623656e+002 -2.4031564e+002 -3.7257428e+002 -3.4108233e+002 $\Delta w_4$ -3.0865547e+002 -2.8842569e+002 -2.9824681e+002 -2.6018254e+002 -2.7533841e+002 $\Delta w_5$ 7.0391477e+002 6.9146259e+002 6.8271055e+002 6.6623463e+002 6.5626173e+002 $\Delta w_6$ -2.7364462e-002 -3.5663105e-002 -4.9939414e-002 1.0528198e-003 -3.5473078e-001 $\Delta w_7$ -1.3859574e+002 -1.1951280e+002 -2.1609022e+002 -1.0360357e+002 -1.0937146e+002 $\Delta w_8$ -1.1308908e-001 -2.4663699e-003 -2.2421419e-003 -7.7830259e-003 -1.7166429e-007 $\Delta w_9$ 2.1891752e-003 1.6847602e-007 5.8640523e-002 3.0980424e-003 7.3055641e-001 $\Delta w_{10}$ -2.8203111e-007 -3.2609527e-004 -7.4650111e-008 -1.1673710e-006 -3.2439030e-003
 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ 2.1589291e-002 -3.4351365e-002 5.7805964e-002 2.3686226e-003 1.0158904e-003 $\Delta w_2$ -2.8283011e-001 -2.7323758e-004 1.6033220e-003 -3.5364786e-003 -2.6401259e-004 $\Delta w_3$ -3.3036266e+002 -3.5623656e+002 -2.4031564e+002 -3.7257428e+002 -3.4108233e+002 $\Delta w_4$ -3.0865547e+002 -2.8842569e+002 -2.9824681e+002 -2.6018254e+002 -2.7533841e+002 $\Delta w_5$ 7.0391477e+002 6.9146259e+002 6.8271055e+002 6.6623463e+002 6.5626173e+002 $\Delta w_6$ -2.7364462e-002 -3.5663105e-002 -4.9939414e-002 1.0528198e-003 -3.5473078e-001 $\Delta w_7$ -1.3859574e+002 -1.1951280e+002 -2.1609022e+002 -1.0360357e+002 -1.0937146e+002 $\Delta w_8$ -1.1308908e-001 -2.4663699e-003 -2.2421419e-003 -7.7830259e-003 -1.7166429e-007 $\Delta w_9$ 2.1891752e-003 1.6847602e-007 5.8640523e-002 3.0980424e-003 7.3055641e-001 $\Delta w_{10}$ -2.8203111e-007 -3.2609527e-004 -7.4650111e-008 -1.1673710e-006 -3.2439030e-003
Solution with five period; $\varsigma = 0.01, \delta = 0.1$
 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ -5.4467604e-002 -7.0948978e-007 -2.2561076e-004 -1.0737564e-002 -1.6291721e-004 $\Delta w_2$ -1.6419802e-003 -5.6357371e-007 -7.4291123e-003 4.3365254e-003 -2.9140832e-004 $\Delta w_3$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_4$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_5$ 1.7420318e+003 1.7397785e+003 1.7370980e+003 1.7349762e+003 1.7353447e+003 $\Delta w_6$ -9.7249593e-008 -3.9907242e-002 -1.8502745e-007 -4.0113434e-002 -3.8173914e-004 $\Delta w_7$ -7.9037071e+002 -7.9581203e+002 -7.9336516e+002 -7.8689932e+002 -7.9102920e+002 $\Delta w_8$ -2.1545195e+002 -2.0339581e+002 -2.0451219e+002 -2.0476930e+002 -1.9763968e+002 $\Delta w_9$ 1.2043583e+003 1.2000782e+003 1.2014231e+003 1.1974970e+003 1.1941434e+003 $\Delta w_{10}$ -3.4352764e-002 -2.3689203e-007 -1.2659914e-004 -1.2481443e-007 -2.0840295e-007
 $k_1$ $k_2$ $k_3$ $k_4$ $k_5$ $\Delta w_1$ -5.4467604e-002 -7.0948978e-007 -2.2561076e-004 -1.0737564e-002 -1.6291721e-004 $\Delta w_2$ -1.6419802e-003 -5.6357371e-007 -7.4291123e-003 4.3365254e-003 -2.9140832e-004 $\Delta w_3$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_4$ -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 -1.0000000e+003 $\Delta w_5$ 1.7420318e+003 1.7397785e+003 1.7370980e+003 1.7349762e+003 1.7353447e+003 $\Delta w_6$ -9.7249593e-008 -3.9907242e-002 -1.8502745e-007 -4.0113434e-002 -3.8173914e-004 $\Delta w_7$ -7.9037071e+002 -7.9581203e+002 -7.9336516e+002 -7.8689932e+002 -7.9102920e+002 $\Delta w_8$ -2.1545195e+002 -2.0339581e+002 -2.0451219e+002 -2.0476930e+002 -1.9763968e+002 $\Delta w_9$ 1.2043583e+003 1.2000782e+003 1.2014231e+003 1.1974970e+003 1.1941434e+003 $\Delta w_{10}$ -3.4352764e-002 -2.3689203e-007 -1.2659914e-004 -1.2481443e-007 -2.0840295e-007

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