American Institute of Mathematical Sciences

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

Robust multi-period and multi-objective portfolio selection

Lin Jiang , and  Song Wang

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  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.

Keywords: Portfolio selection, robust optimization, multi-objective optimization, weighted-sum.
Mathematics Subject Classification: Primary: 90C26, 90C29, 65K05; Secondary: 91G10.
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
References:
[1]

A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Math. Program., 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar

[2]

X. Cui, X. Sun and D. Sha, An empirical study on discrete optimization models for portfolio selection, J. Ind. Manag. Optim., 5 (2009), 33-46.  doi: 10.3934/jimo.2009.5.33.  Google Scholar

[3]

K. Deb, A. Pratap, S. Agarwal and T. A. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Comput., 6 (2002), 182-197.  doi: 10.1109/4235.996017.  Google Scholar

[4]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res., 28 (2003), 1-38.  doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[5]

D. Huang, S. Zhu, F. J. Fabozzi and M. Fukushima, Portfolio selection under distributional uncertainty: A relative robust CVaR approach, European J. Oper. Res., 203 (2010), 185-194.  doi: 10.1016/j.ejor.2009.07.010.  Google Scholar

[6]

K. Khalili-Damghani and M. Amiri, Solving binary-state multi-objective reliability redundancy allocation series-parallel problem using efficient epsilon-constraint, multi-start partial bound enumeration algorithm, and DEA, Reliability Engineering and System Safety, 103 (2012), 35-44.  doi: 10.1016/j.ress.2012.03.006.  Google Scholar

[7]

C. C. Lin and Y. T. Liu, Genetic algorithms for portfolio selection problems with minimum transaction lots, European J. Oper. Res., 185 (2008), 393-404.  doi: 10.1016/j.ejor.2006.12.024.  Google Scholar

[8]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, Inc., New York, 1959.  Google Scholar

[9]

R. T. Marler and J. S. Arora, The weighted sum method for multi-objective optimization: New insights, Struct. Multidiscip. Optim., 41 (2010), 853-862.  doi: 10.1007/s00158-009-0460-7.  Google Scholar

[10]

K. Ruan and M. Fukushima, Robust portfolio selection with a combined WCVaR and factor model, J. Ind. Manag. Optim., 8 (2012), 343-362.  doi: 10.3934/jimo.2012.8.343.  Google Scholar

[11]

K. Schottle and R. Werner, Robustness properties of mean-variance portfolios, Optimization, 58 (2009), 641-663.  doi: 10.1080/02331930902819220.  Google Scholar

[12]

H. Soleimani, H. R. Golmakani and M. H. Salimi, Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm, Expert Systems with Appl., 36 (2009), 5058-5063.  doi: 10.1016/j.eswa.2008.06.007.  Google Scholar

[13]

Q. Wang and H. Sun, Sparse Markowitz portfolio selection by using stochastic linear complementarity approach, J. Ind. Manag. Optim., 14 (2018), 541-559.  doi: 10.3934/jimo.2017059.  Google Scholar

[14]

Z. Wang and S. Liu, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs, J. Ind. Manag. Optim., 9 (2013), 643-657.  doi: 10.3934/jimo.2013.9.643.  Google Scholar

[15]

C. Wu, K. L. Teo and S. Wu, Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica J. IFAC, 49 (2013), 1809-1815.  doi: 10.1016/j.automatica.2013.02.052.  Google Scholar

[16]

P. Xidonas, G. Mavrotas, C. Hassapis and C. Zopounidis, Robust multiobjective portfolio optimization: A minimax regret approach, European J. Oper. Res., 262 (2017), 299-305.  doi: 10.1016/j.ejor.2017.03.041.  Google Scholar

[17]

L. Yi, Z. F. Li and D. Li, Multi-period portfolio selection for asset-liability management with uncertain investment horizon, J. Ind. Manag. Optim., 4 (2008), 535-552.  doi: 10.3934/jimo.2008.4.535.  Google Scholar

[18]

N. Zhang, A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems, J. Ind. Manag. Optim., (2018). doi: 10.3934/jimo.2018189.  Google Scholar

[19]

P. Zhang, Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection, J. Ind. Manag. Optim., 15 (2019), 537-564.  doi: 10.3934/jimo.2018056.  Google Scholar

[20]

C. Zhao, C. Wu, J. Chai, X. Wang, X. Yang, J. M. Lee and M. J. Kim, Decomposition-based multi-objective firefly algorithm for RFID network planning with uncertainty, Appl. Soft Comput., 55 (2017), 549-564.  doi: 10.1016/j.asoc.2017.02.009.  Google Scholar

show all references

References:
[1]

A. Ben-Tal and A. Nemirovski, Robust optimization-methodology and applications, Math. Program., 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar

[2]

X. Cui, X. Sun and D. Sha, An empirical study on discrete optimization models for portfolio selection, J. Ind. Manag. Optim., 5 (2009), 33-46.  doi: 10.3934/jimo.2009.5.33.  Google Scholar

[3]

K. Deb, A. Pratap, S. Agarwal and T. A. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Comput., 6 (2002), 182-197.  doi: 10.1109/4235.996017.  Google Scholar

[4]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Math. Oper. Res., 28 (2003), 1-38.  doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[5]

D. Huang, S. Zhu, F. J. Fabozzi and M. Fukushima, Portfolio selection under distributional uncertainty: A relative robust CVaR approach, European J. Oper. Res., 203 (2010), 185-194.  doi: 10.1016/j.ejor.2009.07.010.  Google Scholar

[6]

K. Khalili-Damghani and M. Amiri, Solving binary-state multi-objective reliability redundancy allocation series-parallel problem using efficient epsilon-constraint, multi-start partial bound enumeration algorithm, and DEA, Reliability Engineering and System Safety, 103 (2012), 35-44.  doi: 10.1016/j.ress.2012.03.006.  Google Scholar

[7]

C. C. Lin and Y. T. Liu, Genetic algorithms for portfolio selection problems with minimum transaction lots, European J. Oper. Res., 185 (2008), 393-404.  doi: 10.1016/j.ejor.2006.12.024.  Google Scholar

[8]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, Inc., New York, 1959.  Google Scholar

[9]

R. T. Marler and J. S. Arora, The weighted sum method for multi-objective optimization: New insights, Struct. Multidiscip. Optim., 41 (2010), 853-862.  doi: 10.1007/s00158-009-0460-7.  Google Scholar

[10]

K. Ruan and M. Fukushima, Robust portfolio selection with a combined WCVaR and factor model, J. Ind. Manag. Optim., 8 (2012), 343-362.  doi: 10.3934/jimo.2012.8.343.  Google Scholar

[11]

K. Schottle and R. Werner, Robustness properties of mean-variance portfolios, Optimization, 58 (2009), 641-663.  doi: 10.1080/02331930902819220.  Google Scholar

[12]

H. Soleimani, H. R. Golmakani and M. H. Salimi, Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm, Expert Systems with Appl., 36 (2009), 5058-5063.  doi: 10.1016/j.eswa.2008.06.007.  Google Scholar

[13]

Q. Wang and H. Sun, Sparse Markowitz portfolio selection by using stochastic linear complementarity approach, J. Ind. Manag. Optim., 14 (2018), 541-559.  doi: 10.3934/jimo.2017059.  Google Scholar

[14]

Z. Wang and S. Liu, Multi-period mean-variance portfolio selection with fixed and proportional transaction costs, J. Ind. Manag. Optim., 9 (2013), 643-657.  doi: 10.3934/jimo.2013.9.643.  Google Scholar

[15]

C. Wu, K. L. Teo and S. Wu, Min-max optimal control of linear systems with uncertainty and terminal state constraints, Automatica J. IFAC, 49 (2013), 1809-1815.  doi: 10.1016/j.automatica.2013.02.052.  Google Scholar

[16]

P. Xidonas, G. Mavrotas, C. Hassapis and C. Zopounidis, Robust multiobjective portfolio optimization: A minimax regret approach, European J. Oper. Res., 262 (2017), 299-305.  doi: 10.1016/j.ejor.2017.03.041.  Google Scholar

[17]

L. Yi, Z. F. Li and D. Li, Multi-period portfolio selection for asset-liability management with uncertain investment horizon, J. Ind. Manag. Optim., 4 (2008), 535-552.  doi: 10.3934/jimo.2008.4.535.  Google Scholar

[18]

N. Zhang, A symmetric Gauss-Seidel based method for a class of multi-period mean-variance portfolio selection problems, J. Ind. Manag. Optim., (2018). doi: 10.3934/jimo.2018189.  Google Scholar

[19]

P. Zhang, Chance-constrained multiperiod mean absolute deviation uncertain portfolio selection, J. Ind. Manag. Optim., 15 (2019), 537-564.  doi: 10.3934/jimo.2018056.  Google Scholar

[20]

C. Zhao, C. Wu, J. Chai, X. Wang, X. Yang, J. M. Lee and M. J. Kim, Decomposition-based multi-objective firefly algorithm for RFID network planning with uncertainty, Appl. Soft Comput., 55 (2017), 549-564.  doi: 10.1016/j.asoc.2017.02.009.  Google Scholar

Figure 1.  The variation of $ g(\kappa) $ in terms of $ \kappa $ when $ \lambda = 0.1 $
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Figure 2.  The variation of $ g(\kappa) $ in terms of $ \kappa $ when $ \lambda=0.5 $
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Figure 3.  The variation of $ g(\kappa) $ in terms of $ \kappa $ when $ \lambda = 0.9 $
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Figure 4.  The variation of $ f_{\lambda}(\Delta w_t, \kappa^*) $ in terms of $ \delta $ when $ \lambda = 0.1 $
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Figure 5.  The variation of $ f_{\lambda}(\Delta w_t, \kappa^*) $ in terms of $ \delta $ when $ \lambda = 0.5 $
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Figure 6.  The variation of $ f_{\lambda}(\Delta w_t, \kappa^*) $ in terms of $ \delta $ when $ \lambda = 0.9 $
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Figure 7.  The variation of $ f_{\lambda}(\Delta w_t, \kappa^*) $ in terms of $ \varsigma $ when $ \lambda = 0.5 $
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Figure 8.  The pareton front with $ \delta = 1 $
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Figure 9.  The pareton front with $ \delta = 5 $
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Figure 10.  The pareton front with $ \delta = 10 $
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Table 1.  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
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$ 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
Table 2.  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
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$ 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
Table 3.  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
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$ 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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