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November  2020, 16(6): 2923-2942. doi: 10.3934/jimo.2019087

Distributionally robust chance constrained problems under general moments information

Ke-Wei Ding 1,3, , Nan-Jing Huang 2,, and  Yi-Bin Xiao 3,

1. 

School of Computer Science and Technology, Southwest Minzu University, Chengdu, Sichuan 610041, China
2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
3. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

* Corresponding author

Received  December 2018 Revised  March 2019 Published  November 2020 Early access  July 2019

In this paper, we focus on distributionally robust chance constrained problems (DRCCPs) under general moments information sets. By convex analysis, we obtain an equivalent convex programming form for DRCCP under assumptions that the first and second order moments belong to corresponding convex and compact sets respectively. We give some examples of support functions about matrix sets to show the tractability of the equivalent convex programming and obtain the closed form solution for the worst case VaR optimization problem. Then, we present an equivalent convex programming form for DRCCP under assumptions that the first order moment set and the support subsets are convex and compact. We also give an equivalent form for distributionally robust nonlinear chance constrained problem under assumptions that the first order moment set and the support set are convex and compact. Moreover, we provide illustrative examples to show our results.

Keywords: Distributionally robust, chance constraint, moments uncertainty, support function.
Mathematics Subject Classification: 90C15, 90C25.
Citation: Ke-Wei Ding, Nan-Jing Huang, Yi-Bin Xiao. Distributionally robust chance constrained problems under general moments information. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2923-2942. doi: 10.3934/jimo.2019087
References:
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V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer, New York, 2012. doi: 10.1007/978-94-007-2247-7.

[2]

A. Ben-Tal, D. Bertsimas and D. Brown, A soft robust model for optimization under ambiguity, Operations Research, 58 (2010), 1220-1234.  doi: 10.1287/opre.1100.0821.

[3]

A. Ben-Tal, D. Hertog and J. Vial, Deriving robust counterparts of nonlinear uncertain inequalities, Mathematical Programming, 149 (2015), 265-299.  doi: 10.1007/s10107-014-0750-8.

[4] D. S. Bernstein, Matrix Mathematics, Princeton University Press, New Jersey, 2009.  doi: 10.1515/9781400833344.
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G. Calafiore and L. El Ghaoui, On distributionally robust chance-constrained linear programms with applications, Journal of Optimization Theory and Applications, 130 (2006), 1-22.  doi: 10.1007/s10957-006-9084-x.

[6]

E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.

[7]

K. W. Ding, M. H. Wang and N. J. Huang, Distributionally robust chance constrained problem under interval distribution information, Optimization Letters, 12 (2018), 1315-1328.  doi: 10.1007/s11590-017-1160-7.

[8]

L. Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Operations Research, 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101.

[9]

R. Hu, Y.-B. Xiao, N.-J. Huang and X. Wang, Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459. 

[10]

K. Isii, On sharpness of Tchebychev-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962), 185-197.  doi: 10.1007/BF02868641.

[11]

B. Li, J. Sun, H. Xu and M. Zhang, A class of two-stage distributionally robust stochastic games, Journal of Industrial and Management Optimization, 15 (2019), 387-400. 

[12]

B. Li, X. Qian, J. Sun, K. L. Teo and C. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039.

[13]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.

[14]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109.  doi: 10.1109/LSP.2017.2773601.

[15]

J. Lu, Y.-B. Xiao and N.-J. Huang, A Stackelberg quasi-equilibrium problem via quasi-variational inequalities, Carpathian Journal of Mathematics, 34 (2018), 355-362. 

[16]

W. Li, Y.-B. Xiao, N.-J. Huang and Y. J. Cho, A class of differential inverse quasi-variational inequalities in finite dimensional spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), 4532-4543.  doi: 10.22436/jnsa.010.08.45.

[17]

K. Natarajan, M. Sim and J. Uichanco, Tractable robust expected utility and risk models for portofolio optimization, Mathematical Finance, 20 (2010), 695-731.  doi: 10.1111/j.1467-9965.2010.00417.x.

[18]

A. Petrusel, G. Petrusel, Y.-B. Xiao and J.-C. Yao, Fixed point theorems for generalized contractions with applications to coupled fixed point theory, Journal of Nonlinear and Convex Analysis, 19 (2018), 71-87. 

[19]

I. Pólik and T. Terlaky, A survey of the $\mathcal{S}$-lemma, SIAM Review, 49 (2007), 371-481.  doi: 10.1137/S003614450444614X.

[20]

Q.-Y. Shu, R. Hu and Y.-B. Xiao, Metric characterizations for well-posedness of split hemivariational inequalities, J. Inequal. Appl., (2018), 17 pp. doi: 10.1186/s13660-018-1761-4.

[21]

A. Shapiro and A. Kleywegt, Minimax analysis of stochastic problems, Optimization Methods & Software, 17 (2002), 523-542.  doi: 10.1080/1055678021000034008.

[22]

M. Sofonea, Y.-B. Xiao and M. Couderc, Optimization problems for elastic contact models with unilateral constraints, Z. Angew. Math. Phys., 70 (2019), 17 pp. doi: 10.1007/s00033-018-1046-2.

[23]

M. Sofonea and Y.-B. Xiao, Boundary optimal control of a nonsmooth frictionless contact problem, Comput. Math. Appl., 78 (2019), 152-165.  doi: 10.1016/j.camwa.2019.02.027.

[24]

H. Sun and H. Xu, Convergence analysis for distributionally robust optimization and equilibrium problems, Mathematics of Operations Research, 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.

[25]

X. Tong, H. Sun, X. Luo and Q. Zheng, Distributionally robust chance constrained optimization for economic dispatch in renewable energy integrated systems, Journal of Global Optimization, 70 (2018), 131-158.  doi: 10.1007/s10898-017-0572-3.

[26]

X. Wang, N. Fan and P. Pardalos, Robust chance-constrained support vector machines with second-order moment information, Annals of Operations Research, 263 (2018), 45-68.  doi: 10.1007/s10479-015-2039-6.

[27]

Y.-M. Wang, Y.-B. Xiao, X. Wang and Y. J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.

[28]

W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.

[29]

W. Xie and S. Ahmed, On deterministic reformulations of distributionally robust joint chance constrained optimization problems, SIAM Journal on Optimization, 28 (2018), 1151-1182.  doi: 10.1137/16M1094725.

[30]

Y.-B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, Journal of Mathematical Analysis and Applications, 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.

[31]

Y.-B. Xiao and M. Sofonea, Generalized penalty method for elliptic variational-hemivariational inequalities, Applied Mathematics and Optimization, (2019). doi: 10.1007/s00245-019-09563-4.

[32]

W. Yang and H. Xu, Distributionally robust chance constraints for non-linear uncertainties, Mathematical Programming, 155 (2016), 231-265.  doi: 10.1007/s10107-014-0842-5.

[33]

Y. Zhang, S. Shen and S. Erdogan, Distributionally robust appointment scheduling with moment-based ambiguity set, Operations Research Letters, 45 (2017), 139-144.  doi: 10.1016/j.orl.2017.01.010.

[34]

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.

[35]

S. Zymler, D. Kuhn and B. Rustem, Worst-case value at risk of nonlinear portfolios, Management Science, 59 (2009), 172-188.  doi: 10.1287/mnsc.1120.1615.

show all references

References:
[1]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer, New York, 2012. doi: 10.1007/978-94-007-2247-7.

[2]

A. Ben-Tal, D. Bertsimas and D. Brown, A soft robust model for optimization under ambiguity, Operations Research, 58 (2010), 1220-1234.  doi: 10.1287/opre.1100.0821.

[3]

A. Ben-Tal, D. Hertog and J. Vial, Deriving robust counterparts of nonlinear uncertain inequalities, Mathematical Programming, 149 (2015), 265-299.  doi: 10.1007/s10107-014-0750-8.

[4] D. S. Bernstein, Matrix Mathematics, Princeton University Press, New Jersey, 2009.  doi: 10.1515/9781400833344.
[5]

G. Calafiore and L. El Ghaoui, On distributionally robust chance-constrained linear programms with applications, Journal of Optimization Theory and Applications, 130 (2006), 1-22.  doi: 10.1007/s10957-006-9084-x.

[6]

E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.

[7]

K. W. Ding, M. H. Wang and N. J. Huang, Distributionally robust chance constrained problem under interval distribution information, Optimization Letters, 12 (2018), 1315-1328.  doi: 10.1007/s11590-017-1160-7.

[8]

L. Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Operations Research, 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101.

[9]

R. Hu, Y.-B. Xiao, N.-J. Huang and X. Wang, Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459. 

[10]

K. Isii, On sharpness of Tchebychev-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962), 185-197.  doi: 10.1007/BF02868641.

[11]

B. Li, J. Sun, H. Xu and M. Zhang, A class of two-stage distributionally robust stochastic games, Journal of Industrial and Management Optimization, 15 (2019), 387-400. 

[12]

B. Li, X. Qian, J. Sun, K. L. Teo and C. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Applied Mathematical Modelling, 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039.

[13]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Transactions on Wireless Communications, 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.

[14]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109.  doi: 10.1109/LSP.2017.2773601.

[15]

J. Lu, Y.-B. Xiao and N.-J. Huang, A Stackelberg quasi-equilibrium problem via quasi-variational inequalities, Carpathian Journal of Mathematics, 34 (2018), 355-362. 

[16]

W. Li, Y.-B. Xiao, N.-J. Huang and Y. J. Cho, A class of differential inverse quasi-variational inequalities in finite dimensional spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), 4532-4543.  doi: 10.22436/jnsa.010.08.45.

[17]

K. Natarajan, M. Sim and J. Uichanco, Tractable robust expected utility and risk models for portofolio optimization, Mathematical Finance, 20 (2010), 695-731.  doi: 10.1111/j.1467-9965.2010.00417.x.

[18]

A. Petrusel, G. Petrusel, Y.-B. Xiao and J.-C. Yao, Fixed point theorems for generalized contractions with applications to coupled fixed point theory, Journal of Nonlinear and Convex Analysis, 19 (2018), 71-87. 

[19]

I. Pólik and T. Terlaky, A survey of the $\mathcal{S}$-lemma, SIAM Review, 49 (2007), 371-481.  doi: 10.1137/S003614450444614X.

[20]

Q.-Y. Shu, R. Hu and Y.-B. Xiao, Metric characterizations for well-posedness of split hemivariational inequalities, J. Inequal. Appl., (2018), 17 pp. doi: 10.1186/s13660-018-1761-4.

[21]

A. Shapiro and A. Kleywegt, Minimax analysis of stochastic problems, Optimization Methods & Software, 17 (2002), 523-542.  doi: 10.1080/1055678021000034008.

[22]

M. Sofonea, Y.-B. Xiao and M. Couderc, Optimization problems for elastic contact models with unilateral constraints, Z. Angew. Math. Phys., 70 (2019), 17 pp. doi: 10.1007/s00033-018-1046-2.

[23]

M. Sofonea and Y.-B. Xiao, Boundary optimal control of a nonsmooth frictionless contact problem, Comput. Math. Appl., 78 (2019), 152-165.  doi: 10.1016/j.camwa.2019.02.027.

[24]

H. Sun and H. Xu, Convergence analysis for distributionally robust optimization and equilibrium problems, Mathematics of Operations Research, 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.

[25]

X. Tong, H. Sun, X. Luo and Q. Zheng, Distributionally robust chance constrained optimization for economic dispatch in renewable energy integrated systems, Journal of Global Optimization, 70 (2018), 131-158.  doi: 10.1007/s10898-017-0572-3.

[26]

X. Wang, N. Fan and P. Pardalos, Robust chance-constrained support vector machines with second-order moment information, Annals of Operations Research, 263 (2018), 45-68.  doi: 10.1007/s10479-015-2039-6.

[27]

Y.-M. Wang, Y.-B. Xiao, X. Wang and Y. J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.

[28]

W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.

[29]

W. Xie and S. Ahmed, On deterministic reformulations of distributionally robust joint chance constrained optimization problems, SIAM Journal on Optimization, 28 (2018), 1151-1182.  doi: 10.1137/16M1094725.

[30]

Y.-B. Xiao and M. Sofonea, On the optimal control of variational-hemivariational inequalities, Journal of Mathematical Analysis and Applications, 475 (2019), 364-384.  doi: 10.1016/j.jmaa.2019.02.046.

[31]

Y.-B. Xiao and M. Sofonea, Generalized penalty method for elliptic variational-hemivariational inequalities, Applied Mathematics and Optimization, (2019). doi: 10.1007/s00245-019-09563-4.

[32]

W. Yang and H. Xu, Distributionally robust chance constraints for non-linear uncertainties, Mathematical Programming, 155 (2016), 231-265.  doi: 10.1007/s10107-014-0842-5.

[33]

Y. Zhang, S. Shen and S. Erdogan, Distributionally robust appointment scheduling with moment-based ambiguity set, Operations Research Letters, 45 (2017), 139-144.  doi: 10.1016/j.orl.2017.01.010.

[34]

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.

[35]

S. Zymler, D. Kuhn and B. Rustem, Worst-case value at risk of nonlinear portfolios, Management Science, 59 (2009), 172-188.  doi: 10.1287/mnsc.1120.1615.

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