具有二次目标函数的多阶段随机规划问题的稳定性研究（英文）
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摘要
多阶段随机规划可恰当描述不确定环境下的复杂长期决策问题.本文研究带有二次目标函数的多阶段随机规划问题在随机过程扰动下的定量稳定性,推广了现有线性目标函数情形下的结果.为此,我们首先根据参数规划的相关理论导出了可行解的上界.为了得到补偿函数的Lipschitz连续性,我们假设了Fortet-Mourier度量下随机过程各阶段条件分布下的连续性.在这些准备工作的基础上,我们最终建立了最优值函数的Lipschitz连续性结论.我们的定量稳定性结果推广了已有的线性结果,并不依赖于多阶段随机规划稳定性分析中难以计算的滤波距离.
Multistage stochastic programs can properly describe complex long-term decisionmaking problems under uncertainty. We study the quantitative stability of multistage stochastic programs with quadratic objective functions under perturbations of the underlying stochastic processes, which extend the current results for the linear objective functions. We first derive the upper bounds of feasible solutions through parametric programming theories. In order to obtain the Lipschitz continuities of recourse functions, we assume the continuity of the conditional distributions under the Fortet-Mourier metric. With these preparations, we finally establish the Lipschitz continuity of the optimal value function. Our quantitative stability results do not rely on the filtration distance.
Multistage stochastic programs can properly describe complex long-term decisionmaking problems under uncertainty. We study the quantitative stability of multistage stochastic programs with quadratic objective functions under perturbations of the underlying stochastic processes, which extend the current results for the linear objective functions. We first derive the upper bounds of feasible solutions through parametric programming theories. In order to obtain the Lipschitz continuities of recourse functions, we assume the continuity of the conditional distributions under the Fortet-Mourier metric. With these preparations, we finally establish the Lipschitz continuity of the optimal value function. Our quantitative stability results do not rely on the filtration distance.
引文
[1] Pflug G C, Pichler A. Multistage Stochastic Optimization[M]. Cham:Springer, 2014
[2] Kuhn D. Generalized Bounds for Convex Multistage Stochastic Programs[M]. Heidelberg:Springer, 2006
[3] Shapiro A, Dentcheva D, Ruszczynski A. Lectures on Stochastic Programming:Modeling and Theory[M].Philadelphia:SIAM, 2014
[4] Heitsch H, Romisch W, Strugarek C. Stability of multistage stochastic programs[J]. SIAM Journal on Control and Optimization, 2006, 17(2):511-525
[5] Kuchler C. On stability of multistage stochastic programs[J]. SIAM Journal on Control and Optimization,2008, 19(2):952-968
[6] Eichhorn A, Romisch W. Stability of multistage stochastic programs incorporating polyhedral risk measures[J]. Optimization, 2008, 57(2):295-318
[7] Heitsch H, Romisch W. Scenario tree modeling for multistage stochastic programs[J]. Mathematical Programming, 2009, 118(2):371-406
[8] Mirkov R, Pflug G C. Tree approximations of dynamic stochastic programs[J]. SIAM Journal on Control and Optimization, 2007, 18(3):1082-1105
[9] Fiedler O, Romisch W. Stability in multistage stochastic programming[J]. Annals of Operations Research,1995, 56(1):79-93
[10] Shapiro A. Inference of statistical bounds for multistage stochastic programming problems[J]. Mathematical Methods of Operations Research, 2003, 58(1):57-68
[11] Pennanen T. Epi-convergent discretizations of multistage stochastic programs via integration quadratures[J]. Mathematical Programming, 2009, 116(1):461-479
[12] Eichhorn A, Romisch W. Polyhedral risk measures in stochastic programming[J]. SIAM Journal on Control and Optimization, 2005, 16(1):69-95
[13] Guigues V, Romisch W. Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures[J]. SIAM Journal on Control and Optimization, 2012, 22(2):286-312
[14] Philpott A, de Matos V, Finardi E. On solving multistage stochastic programs with coherent risk measures[J]. Operations Research, 2013, 61(4):957-970
[15] Dupacova J, Kozm?k V. Structure of risk-averse multistage stochastic programs[J]. OR Spectrum, 2015,37(3):559-582
[16] Homem-de-Mello T, Pagnoncelli B K. Risk aversion in multistage stochastic programming:a modeling and algorithmic perspective[J]. European Journal of Operational Research, 2016, 249(1):188-199
[17] Romisch W, Vigerske S. Quantitative stability of fully random mixed-integer two-stage stochastic programs[J]. Optimization Letters, 2008, 2(3):377-388
[18] Chen Z P, Han Y P. Quantitative stability of mixed-integer two-stage quadratic stochastic programs[J].Mathematical Methods of Operations Research, 2012, 75(2):149-163
[19] Noyan N, Merakli M, Kucukyavuz S. Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design[J]. To preprint on arXiv, 2017
[20] Rockafellar R T, Wets R J B. Variational Analysis[M]. Berlin:Springer, 2009
[21] Han Y P, Chen Z P. Quantitative stability of full random two-stage stochastic programs with recourse[J].Optimization Letters, 2015, 9(6):1075-1090
[22] Reemtsen R, Ruckmann J J. Semi-infinite Programming[M]. Berlin:Springer, 1998
[23] Romisch W. Stability of stochastic programming problems[J]. Handbooks in Operations Research and Management Science, 2003, 10:483-554
[24] Evstigneev I V. Measurable selection and dynamic programming[J]. Mathematics of Operations Research,1976, 1(3):267-272
[2] Kuhn D. Generalized Bounds for Convex Multistage Stochastic Programs[M]. Heidelberg:Springer, 2006
[3] Shapiro A, Dentcheva D, Ruszczynski A. Lectures on Stochastic Programming:Modeling and Theory[M].Philadelphia:SIAM, 2014
[4] Heitsch H, Romisch W, Strugarek C. Stability of multistage stochastic programs[J]. SIAM Journal on Control and Optimization, 2006, 17(2):511-525
[5] Kuchler C. On stability of multistage stochastic programs[J]. SIAM Journal on Control and Optimization,2008, 19(2):952-968
[6] Eichhorn A, Romisch W. Stability of multistage stochastic programs incorporating polyhedral risk measures[J]. Optimization, 2008, 57(2):295-318
[7] Heitsch H, Romisch W. Scenario tree modeling for multistage stochastic programs[J]. Mathematical Programming, 2009, 118(2):371-406
[8] Mirkov R, Pflug G C. Tree approximations of dynamic stochastic programs[J]. SIAM Journal on Control and Optimization, 2007, 18(3):1082-1105
[9] Fiedler O, Romisch W. Stability in multistage stochastic programming[J]. Annals of Operations Research,1995, 56(1):79-93
[10] Shapiro A. Inference of statistical bounds for multistage stochastic programming problems[J]. Mathematical Methods of Operations Research, 2003, 58(1):57-68
[11] Pennanen T. Epi-convergent discretizations of multistage stochastic programs via integration quadratures[J]. Mathematical Programming, 2009, 116(1):461-479
[12] Eichhorn A, Romisch W. Polyhedral risk measures in stochastic programming[J]. SIAM Journal on Control and Optimization, 2005, 16(1):69-95
[13] Guigues V, Romisch W. Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures[J]. SIAM Journal on Control and Optimization, 2012, 22(2):286-312
[14] Philpott A, de Matos V, Finardi E. On solving multistage stochastic programs with coherent risk measures[J]. Operations Research, 2013, 61(4):957-970
[15] Dupacova J, Kozm?k V. Structure of risk-averse multistage stochastic programs[J]. OR Spectrum, 2015,37(3):559-582
[16] Homem-de-Mello T, Pagnoncelli B K. Risk aversion in multistage stochastic programming:a modeling and algorithmic perspective[J]. European Journal of Operational Research, 2016, 249(1):188-199
[17] Romisch W, Vigerske S. Quantitative stability of fully random mixed-integer two-stage stochastic programs[J]. Optimization Letters, 2008, 2(3):377-388
[18] Chen Z P, Han Y P. Quantitative stability of mixed-integer two-stage quadratic stochastic programs[J].Mathematical Methods of Operations Research, 2012, 75(2):149-163
[19] Noyan N, Merakli M, Kucukyavuz S. Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design[J]. To preprint on arXiv, 2017
[20] Rockafellar R T, Wets R J B. Variational Analysis[M]. Berlin:Springer, 2009
[21] Han Y P, Chen Z P. Quantitative stability of full random two-stage stochastic programs with recourse[J].Optimization Letters, 2015, 9(6):1075-1090
[22] Reemtsen R, Ruckmann J J. Semi-infinite Programming[M]. Berlin:Springer, 1998
[23] Romisch W. Stability of stochastic programming problems[J]. Handbooks in Operations Research and Management Science, 2003, 10:483-554
[24] Evstigneev I V. Measurable selection and dynamic programming[J]. Mathematics of Operations Research,1976, 1(3):267-272
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