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August & September  2019, 12(4&5): 761-770. doi: 10.3934/dcdss.2019050

A new approach for worst-case regret portfolio optimization problem

 1 Business School, University of Shanghai for Science and Technology, Shanghai 200093, China 2 School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

* Corresponding author: Shaojian Qu

Received  June 2017 Revised  October 2017 Published  November 2018

Fund Project: The first author is supported by NSF grant 71571055.

This paper considers the worst-case regret portfolio optimization problem when the distributions of the asset returns are uncertain. In general, the solution to this problem is NP hard and approximation methods that minimise the difference between the maximum return and the sum of each portfolio return are often proposed. Applying the duality of semi-infinite programming, the worst-case regret portfolio optimization problem with uncertain distributions can be equivalently reformulated to a linear optimization problem, and the established solution approaches for linear optimization can then be applied. An example of a portfolio optimization problem is provided to show the efficiency of our method and the results demonstrate that our method can satisfy the portfolio risk diversification property under the uncertain distributions of the returns.

Citation: Ying Ji, Shaojian Qu, Yeming Dai. A new approach for worst-case regret portfolio optimization problem. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 761-770. doi: 10.3934/dcdss.2019050
References:
 [1] A. Ben Tal and A. Nemirovski, Robust convex optimizaion, Math Oper Res, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769. [2] L. Chen, S. He and S. Zhang, Tight bounds for some risk measures with applications to robust portfolio selection, Operations Research, 59 (2011), 847-855.  doi: 10.1287/opre.1110.0950. [3] X. Chen, M. Sim and P. Sun, A robust optimization perspective on stochastic programming, Operational Research, 55 (2007), 1058-1071.  doi: 10.1287/opre.1070.0441. [4] E. Delage and Y. Y. Ye, Distributionally robust optimization under moment uncertainty with application data-driven problems, Operational Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741. [5] J. Goh and M. Sim, Distributionally robust optimization and its tractable approximations, Operational Research, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795. [6] K. Isii, On the sharpness of Chebyshev-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962/1963), 185-197.  doi: 10.1007/BF02868641. [7] Y. Ji, T. N. Wang and M. Goh, The worst-case discounted regret portfolio optimization problem, Applied Mathematics and Computation, 239 (2014), 310-319.  doi: 10.1016/j.amc.2014.04.072. [8] X. Li, B. Y. Shou and Z. F. Qin, An expected regret minimization portfolio selection model, European Journal of Operational Research, 218 (2012), 484-492.  doi: 10.1016/j.ejor.2011.11.015. [9] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [10] D. Y. Meng, Q. Zhao and Z. B. Xu, Improve robustness of sparse PCA by $L_1$-norm maximaization, Pattern Recognit, 45 (2012), 487-497. [11] X. J. Tong and F. Wu, Robust reward-risk ratio optimization with application in allocation of generation asset, Optimization, 63 (2014), 1761-1779.  doi: 10.1080/02331934.2012.672419. [12] M. R. Wagner, Fully distribution-free profit maximization: The inventory management case, Math Operation Reserch, 35 (2010), 728-741.  doi: 10.1287/moor.1100.0468. [13] W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314. [14] S. Zymler, D. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.

show all references

References:
 [1] A. Ben Tal and A. Nemirovski, Robust convex optimizaion, Math Oper Res, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769. [2] L. Chen, S. He and S. Zhang, Tight bounds for some risk measures with applications to robust portfolio selection, Operations Research, 59 (2011), 847-855.  doi: 10.1287/opre.1110.0950. [3] X. Chen, M. Sim and P. Sun, A robust optimization perspective on stochastic programming, Operational Research, 55 (2007), 1058-1071.  doi: 10.1287/opre.1070.0441. [4] E. Delage and Y. Y. Ye, Distributionally robust optimization under moment uncertainty with application data-driven problems, Operational Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741. [5] J. Goh and M. Sim, Distributionally robust optimization and its tractable approximations, Operational Research, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795. [6] K. Isii, On the sharpness of Chebyshev-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962/1963), 185-197.  doi: 10.1007/BF02868641. [7] Y. Ji, T. N. Wang and M. Goh, The worst-case discounted regret portfolio optimization problem, Applied Mathematics and Computation, 239 (2014), 310-319.  doi: 10.1016/j.amc.2014.04.072. [8] X. Li, B. Y. Shou and Z. F. Qin, An expected regret minimization portfolio selection model, European Journal of Operational Research, 218 (2012), 484-492.  doi: 10.1016/j.ejor.2011.11.015. [9] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [10] D. Y. Meng, Q. Zhao and Z. B. Xu, Improve robustness of sparse PCA by $L_1$-norm maximaization, Pattern Recognit, 45 (2012), 487-497. [11] X. J. Tong and F. Wu, Robust reward-risk ratio optimization with application in allocation of generation asset, Optimization, 63 (2014), 1761-1779.  doi: 10.1080/02331934.2012.672419. [12] M. R. Wagner, Fully distribution-free profit maximization: The inventory management case, Math Operation Reserch, 35 (2010), 728-741.  doi: 10.1287/moor.1100.0468. [13] W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314. [14] S. Zymler, D. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.
Numerical Results
 This paper [4] Ix $D^1$ $D^2$ Iter Time Wrv Time Iter Wdrv Ix 50 10 6 0.012 0.2024 0.017 7 0.2255 0.1024 50 25 6 0.013 0.2133 0.019 7 0.2371 0.1003 100 25 21 0.11 0.2978 0.174 27 0.2494 -0.1941 100 50 23 0.129 0.2347 0.285 42 0.2936 0.1914 200 50 37 0.930 0.5484 1.708 46 0.5195 -0.0556 200 100 37 1.288 0.3922 2.502 37 0.3184 -0.2318 300 100 49 3.577 0.5266 4.81 59 0.7026 0.2505 300 200 78 9.563 0.4486 11.84 81 0.4887 0.0821
 This paper [4] Ix $D^1$ $D^2$ Iter Time Wrv Time Iter Wdrv Ix 50 10 6 0.012 0.2024 0.017 7 0.2255 0.1024 50 25 6 0.013 0.2133 0.019 7 0.2371 0.1003 100 25 21 0.11 0.2978 0.174 27 0.2494 -0.1941 100 50 23 0.129 0.2347 0.285 42 0.2936 0.1914 200 50 37 0.930 0.5484 1.708 46 0.5195 -0.0556 200 100 37 1.288 0.3922 2.502 37 0.3184 -0.2318 300 100 49 3.577 0.5266 4.81 59 0.7026 0.2505 300 200 78 9.563 0.4486 11.84 81 0.4887 0.0821
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