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A chance-constrained stochastic model predictive control problem with disturbance feedback
Robust stochastic optimization with convex risk measures: A discretized subgradient scheme
1. | School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Fiance, China |
2. | School of Mathematical Science, Chongqing Normal University, China |
3. | Faculty of Science and Engineering, Curtin University, Perth, Australia |
We study the distributionally robust stochastic optimization problem within a general framework of risk measures, in which the ambiguity set is described by a spectrum of practically used probability distribution constraints such as bounds on mean-deviation and entropic value-at-risk. We show that a subgradient of the objective function can be obtained by solving a finite-dimensional optimization problem, which facilitates subgradient-type algorithms for solving the robust stochastic optimization problem. We develop an algorithm for two-stage robust stochastic programming with conditional value at risk measure. A numerical example is presented to show the effectiveness of the proposed method.
References:
[1] |
A. Ahmadi-Javid,
Entropic value-at-risk: A new coherent risk measure, J. Optim. Theory Appl., 155 (2012), 1105-1123.
doi: 10.1007/s10957-011-9968-2. |
[2] |
J. Ang, F. Meng and J. Sun,
Two-stage stochastic linear programs with incomplete information on uncertainty, European J. Oper. Res., 233 (2014), 16-22.
doi: 10.1016/j.ejor.2013.07.039. |
[3] |
M. Ang, J. Sun and Q. Yao,
On the dual representation of coherent risk measures, Ann. Oper. Res., 262 (2018), 29-46.
doi: 10.1007/s10479-017-2441-3. |
[4] |
D. P. Bertsekas, Convex optimization algorithms, Athena Scientific, Belmont, MA, 2015. |
[5] |
D. P. Bertsekas, A. Nedi and A. E. Ozdaglar, Convex analysis and optimization, Athena Scientific, Belmont, MA, 2003. |
[6] |
D. Bertsimas, X. V. Doan and K. Natarajan,
Models for minimax stochastic linear optimization problems with risk aversion, Math. Oper. Res., 35 (2010), 580-602.
doi: 10.1287/moor.1100.0445. |
[7] |
D. Bertsimas and R. Freund, Data, Models, and Decisions: The Fundamentals of Management Science, South-Western College Publishing, Cincinnati, OH, 2000. Google Scholar |
[8] |
D. Bertsimas, M. Sim and M. Zhang, Adaptive distributionally robust optimization, Manag. Sci., (2018).
doi: 10.1287/mnsc.2017.2952. |
[9] |
G. C. Calalore,
Ambiguous risk measures and optimal robust portfolios, SIAM J. Optim., 18 (2007), 853-877.
doi: 10.1137/060654803. |
[10] |
E. Delage and Y. Y. Ye,
Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res., 58 (2010), 595-612.
doi: 10.1287/opre.1090.0741. |
[11] |
H. Föllmer and A. Schied, Stochastic finance, Walter de Gruyter & Co., Berlin, 2002.
doi: 10.1515/9783110198065. |
[12] |
S. Gao, L. Kong and J. Sun,
Robust two-stage stochastic linear programs with moment constraints, Optimization, 63 (2014), 829-837.
doi: 10.1080/02331934.2014.906598. |
[13] |
M. Grötschel, L. Lovász and and A. Schrijver,
The ellipsoid method and its consequences in combinatorial optimization, Combinatorica, 1 (1981), 169-197.
doi: 10.1007/BF02579273. |
[14] |
Z. Hu and J. Hong, Kullback-Leibler divergence constrained distributionally robust optimization, Available from: http://www.optimization-online.org/DB_HTML/2012/11/3677.html. Google Scholar |
[15] |
D. Klabjan, D. Simchi-Levi and M. Song,
Robust stochastic lot-sizing by means of histograms, Prod. Oper. Manag., 22 (2013), 691-710.
doi: 10.1111/j.1937-5956.2012.01420.x. |
[16] |
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, Appl. Math. Model., 58 (2018), 86-97.
doi: 10.1016/j.apm.2017.11.039. |
[17] |
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 Trans. Signal Process., 16 (2017), 464-474.
doi: 10.1109/TWC.2016.2625246. |
[18] |
B. Li, Y. Rong, J. Sun and K. L. Teo,
A distributionally robust minimum variance beamformer design, IEEE Signal Process. Lett., 25 (2018), 105-109.
doi: 10.1109/LSP.2017.2773601. |
[19] |
B. Li, J. Sun, H. L. Xu and M. Zhang,
A class of two-stage distributionally robust games, J. Ind. Manag. Optim., 15 (2019), 387-400.
doi: 10.3934/jimo.2018048. |
[20] |
A. Ling, J. Sun and X. Yang,
Robust tracking error portfolio selection with worst-case downside risk measures, J. Econom. Dynam. Control, 39 (2014), 178-207.
doi: 10.1016/j.jedc.2013.11.011. |
[21] |
A. Ling, J. Sun, N. H. Xiu and X. Yang,
Robust two-stage stochastic linear optimization with risk aversion, European J. Oper. Res., 256 (2017), 215-229.
doi: 10.1016/j.ejor.2016.06.017. |
[22] |
H-J. Lüthi and J. Doege,
Convex risk measures for portfolio optimization and concepts of flexibility, Math. Program., 104 (2005), 541-559.
doi: 10.1007/s10107-005-0628-x. |
[23] |
S. Mehrotra and H. Zhang,
Models and algorithms for distributionally robust least squares problems, Math. Program., 146 (2014), 123-141.
doi: 10.1007/s10107-013-0681-9. |
[24] |
R. T. Rockafellar, Coherent approaches to risk in optimization under uncertainty, Tutorials in Operations Research, INFORMS, 2007, 38–61.
doi: 10.1287/educ.1073.0032. |
[25] |
R. T. Rockafellar and S. Uryasev,
Optimization of conditional value-at-risk, J. Risk., 2 (2000), 21-42.
doi: 10.21314/JOR.2000.038. |
[26] |
R. T. Rockafellar and S. Uryasev,
Conditional value-at-risk for general loss distributions, J. Banking & Finance, 26 (2002), 1443-1471.
doi: 10.1016/S0378-4266(02)00271-6. |
[27] |
W. W. Rogosinski,
Moments of non-negative mass, Proc. Roy. Soc. London Ser. A, 245 (1958), 1-27.
doi: 10.1098/rspa.1958.0062. |
[28] |
A. Shapiro and S. Ahmed,
On a class of minimax stochastic programs, SIAM J. Optim., 14 (2004), 1237-1249.
doi: 10.1137/S1052623403434012. |
[29] |
A. Shapiro, D. Dentcheva and A. Ruszczyski, Lectures on Stochastic Programming: Modeling and Theory, MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA, 2009.
doi: 10.1137/1.9780898718751. |
[30] |
M. Sion,
On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.
doi: 10.2140/pjm.1958.8.171. |
[31] |
J. Sun, L. Liao and B. Rodrigues,
Quadratic two-stage stochastic optimization with coherent measures of risk, Math. Program., 168 (2018), 599-613.
doi: 10.1007/s10107-017-1131-x. |
[32] |
W. Wiesemann, D. Kuhn and M. Sim,
Distributionally robust convex optimization, Oper. Res., 62 (2014), 1358-1376.
doi: 10.1287/opre.2014.1314. |
show all references
References:
[1] |
A. Ahmadi-Javid,
Entropic value-at-risk: A new coherent risk measure, J. Optim. Theory Appl., 155 (2012), 1105-1123.
doi: 10.1007/s10957-011-9968-2. |
[2] |
J. Ang, F. Meng and J. Sun,
Two-stage stochastic linear programs with incomplete information on uncertainty, European J. Oper. Res., 233 (2014), 16-22.
doi: 10.1016/j.ejor.2013.07.039. |
[3] |
M. Ang, J. Sun and Q. Yao,
On the dual representation of coherent risk measures, Ann. Oper. Res., 262 (2018), 29-46.
doi: 10.1007/s10479-017-2441-3. |
[4] |
D. P. Bertsekas, Convex optimization algorithms, Athena Scientific, Belmont, MA, 2015. |
[5] |
D. P. Bertsekas, A. Nedi and A. E. Ozdaglar, Convex analysis and optimization, Athena Scientific, Belmont, MA, 2003. |
[6] |
D. Bertsimas, X. V. Doan and K. Natarajan,
Models for minimax stochastic linear optimization problems with risk aversion, Math. Oper. Res., 35 (2010), 580-602.
doi: 10.1287/moor.1100.0445. |
[7] |
D. Bertsimas and R. Freund, Data, Models, and Decisions: The Fundamentals of Management Science, South-Western College Publishing, Cincinnati, OH, 2000. Google Scholar |
[8] |
D. Bertsimas, M. Sim and M. Zhang, Adaptive distributionally robust optimization, Manag. Sci., (2018).
doi: 10.1287/mnsc.2017.2952. |
[9] |
G. C. Calalore,
Ambiguous risk measures and optimal robust portfolios, SIAM J. Optim., 18 (2007), 853-877.
doi: 10.1137/060654803. |
[10] |
E. Delage and Y. Y. Ye,
Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res., 58 (2010), 595-612.
doi: 10.1287/opre.1090.0741. |
[11] |
H. Föllmer and A. Schied, Stochastic finance, Walter de Gruyter & Co., Berlin, 2002.
doi: 10.1515/9783110198065. |
[12] |
S. Gao, L. Kong and J. Sun,
Robust two-stage stochastic linear programs with moment constraints, Optimization, 63 (2014), 829-837.
doi: 10.1080/02331934.2014.906598. |
[13] |
M. Grötschel, L. Lovász and and A. Schrijver,
The ellipsoid method and its consequences in combinatorial optimization, Combinatorica, 1 (1981), 169-197.
doi: 10.1007/BF02579273. |
[14] |
Z. Hu and J. Hong, Kullback-Leibler divergence constrained distributionally robust optimization, Available from: http://www.optimization-online.org/DB_HTML/2012/11/3677.html. Google Scholar |
[15] |
D. Klabjan, D. Simchi-Levi and M. Song,
Robust stochastic lot-sizing by means of histograms, Prod. Oper. Manag., 22 (2013), 691-710.
doi: 10.1111/j.1937-5956.2012.01420.x. |
[16] |
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, Appl. Math. Model., 58 (2018), 86-97.
doi: 10.1016/j.apm.2017.11.039. |
[17] |
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 Trans. Signal Process., 16 (2017), 464-474.
doi: 10.1109/TWC.2016.2625246. |
[18] |
B. Li, Y. Rong, J. Sun and K. L. Teo,
A distributionally robust minimum variance beamformer design, IEEE Signal Process. Lett., 25 (2018), 105-109.
doi: 10.1109/LSP.2017.2773601. |
[19] |
B. Li, J. Sun, H. L. Xu and M. Zhang,
A class of two-stage distributionally robust games, J. Ind. Manag. Optim., 15 (2019), 387-400.
doi: 10.3934/jimo.2018048. |
[20] |
A. Ling, J. Sun and X. Yang,
Robust tracking error portfolio selection with worst-case downside risk measures, J. Econom. Dynam. Control, 39 (2014), 178-207.
doi: 10.1016/j.jedc.2013.11.011. |
[21] |
A. Ling, J. Sun, N. H. Xiu and X. Yang,
Robust two-stage stochastic linear optimization with risk aversion, European J. Oper. Res., 256 (2017), 215-229.
doi: 10.1016/j.ejor.2016.06.017. |
[22] |
H-J. Lüthi and J. Doege,
Convex risk measures for portfolio optimization and concepts of flexibility, Math. Program., 104 (2005), 541-559.
doi: 10.1007/s10107-005-0628-x. |
[23] |
S. Mehrotra and H. Zhang,
Models and algorithms for distributionally robust least squares problems, Math. Program., 146 (2014), 123-141.
doi: 10.1007/s10107-013-0681-9. |
[24] |
R. T. Rockafellar, Coherent approaches to risk in optimization under uncertainty, Tutorials in Operations Research, INFORMS, 2007, 38–61.
doi: 10.1287/educ.1073.0032. |
[25] |
R. T. Rockafellar and S. Uryasev,
Optimization of conditional value-at-risk, J. Risk., 2 (2000), 21-42.
doi: 10.21314/JOR.2000.038. |
[26] |
R. T. Rockafellar and S. Uryasev,
Conditional value-at-risk for general loss distributions, J. Banking & Finance, 26 (2002), 1443-1471.
doi: 10.1016/S0378-4266(02)00271-6. |
[27] |
W. W. Rogosinski,
Moments of non-negative mass, Proc. Roy. Soc. London Ser. A, 245 (1958), 1-27.
doi: 10.1098/rspa.1958.0062. |
[28] |
A. Shapiro and S. Ahmed,
On a class of minimax stochastic programs, SIAM J. Optim., 14 (2004), 1237-1249.
doi: 10.1137/S1052623403434012. |
[29] |
A. Shapiro, D. Dentcheva and A. Ruszczyski, Lectures on Stochastic Programming: Modeling and Theory, MPS/SIAM Series on Optimization, SIAM, Philadelphia, PA, 2009.
doi: 10.1137/1.9780898718751. |
[30] |
M. Sion,
On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.
doi: 10.2140/pjm.1958.8.171. |
[31] |
J. Sun, L. Liao and B. Rodrigues,
Quadratic two-stage stochastic optimization with coherent measures of risk, Math. Program., 168 (2018), 599-613.
doi: 10.1007/s10107-017-1131-x. |
[32] |
W. Wiesemann, D. Kuhn and M. Sim,
Distributionally robust convex optimization, Oper. Res., 62 (2014), 1358-1376.
doi: 10.1287/opre.2014.1314. |
1.5 | 1 | |
1 | 2 | |
Molding Machine (hours) | 1 | 1 |
Assembly Machine (hours) | .3 | .5 |
Contribution to Earnings ($/1000 units) | 130 | 100 |
1.5 | 1 | |
1 | 2 | |
Molding Machine (hours) | 1 | 1 |
Assembly Machine (hours) | .3 | .5 |
Contribution to Earnings ($/1000 units) | 130 | 100 |
21000 | 25000 | |
8000 | 0.25 | 0.25 |
10000 | 0.25 | 0.25 |
21000 | 25000 | |
8000 | 0.25 | 0.25 |
10000 | 0.25 | 0.25 |
1 | 2 | 3 | 4 | |
21000 | 21000 | 25000 | 25000 | |
8000 | 10000 | 8000 | 10000 |
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8000 | 10000 | 8000 | 10000 |
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7988 | 9969 | 8000 | 9969 | |
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1 | 2 | 3 | 4 | |
7988 | 9969 | 8000 | 9969 | |
77 | 10 | 27 | 10 |
Pro | 0.5 | 0.0134 | 0.0539 | 0.0006 | 0.4321 |
22355 | 22216 | 21008 | 22239 | 24021 | |
8000 | 10000 | 10000 | 10000 | 10000 |
Pro | 0.5 | 0.0134 | 0.0539 | 0.0006 | 0.4321 |
22355 | 22216 | 21008 | 22239 | 24021 | |
8000 | 10000 | 10000 | 10000 | 10000 |
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