doi: 10.3934/jimo.2020075

Stability of a class of risk-averse multistage stochastic programs and their distributionally robust counterparts

1. 

School of Mathematics and Statistics, Xi’an Jiaotong University, Shaanxi, 710049, China

2. 

Center for Optimization Technique and Quantitative Finance, Xi’an International Academy for Mathematics and Mathematical Technology, Shaanxi, 710049, China

* Corresponding author: Zhiping Chen

Received  February 2019 Revised  January 2020 Published  April 2020

Fund Project: This work is supported by the National Natural Science Foundation of China Grant Numbers 11735011 and 11571270, and the World-Class Universities (Disciplines) and the Characteristic Development Guidance Funds for the Central Universities under Grant Number PY3A058

In this paper, we consider the quantitative stability of a class of risk-averse multistage stochastic programs, whose objective functions are defined by multi-period $ p $th order lower partial moments (LPM) with given targets, and their distributionally robust counterparts. We first derive the upper bounds of feasible solutions as preliminaries. Then, by employing calm modifications, the quantitative stability results are obtained under a special measurable perturbation of stochastic process, which extend the present results under risk-neutral cases to risk-averse ones. Moreover, we recast the risk-averse model by probability measures of stochastic process, and obtain new quantitative stability estimations on the basis of proper probability metrics under the general perturbation of stochastic process. Finally, motivated by the availability of only partial information about probability measures, we further consider the distributionally robust counterpart of our recasting model, and establish the discrepancy of optimal values with respect to the perturbation of ambiguity sets.

Citation: Jie Jiang, Zhiping Chen, He Hu. Stability of a class of risk-averse multistage stochastic programs and their distributionally robust counterparts. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020075
References:
[1]

S. A. Anthonisz, Asset pricing with partial-moments, J. Bank. Financ., 36 (2012), 2122-2135.  doi: 10.1016/j.jbankfin.2012.03.017.  Google Scholar

[2]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics, 28, Princeton University Press, Princeton, NJ, 2009. doi: 10.1515/9781400831050.  Google Scholar

[3]

A. J. Broganab and S. Stidham, Non-separation in the mean–lower-partial-moment portfolio optimization problem, European J. Oper. Res., 184 (2008), 701-710.  doi: 10.1016/j.ejor.2006.11.028.  Google Scholar

[4]

Z. Chen and J. Jiang, Stability analysis of optimization problems with $k$th order stochastic and distributionally robust dominance constraints induced by full random recourse, SIAM J. Optim., 28 (2018), 1396-1419.  doi: 10.1137/17M1120063.  Google Scholar

[5]

J. Dupačová and V. Kozmík, Structure of risk-averse multistage stochastic programs, OR Spectrum, 37 (2015), 559-582.  doi: 10.1007/s00291-014-0379-2.  Google Scholar

[6]

A. Eichhorn and W. Römisch, Polyhedral risk measures in stochastic programming, SIAM J. Optim., 16 (2005), 69-95.  doi: 10.1137/040605217.  Google Scholar

[7]

A. Eichhorn and W. Römisch, Stability of multistage stochastic programs incorporating polyhedral risk measures, Optimization, 57 (2008), 295-318.  doi: 10.1080/02331930701779930.  Google Scholar

[8]

P. C. Fishburn, Mean-risk analysis with risk associated with below-target returns, Amer. Econ. Rev., 67 (1977), 116-126.   Google Scholar

[9]

V. Guigues and W. Römisch, Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures, SIAM J. Optim., 22 (2012), 286-312.  doi: 10.1137/100811696.  Google Scholar

[10]

Y. Han and Z. Chen, Quantitative stability of full random two-stage stochastic programs with recourse, Optim. Lett., 9 (2015), 1075-1090.  doi: 10.1007/s11590-014-0827-6.  Google Scholar

[11]

H. Heitsch and W. Römisch, Scenario tree modeling for multistage stochastic programs, Math. Program., 118 (2009), 371-406.  doi: 10.1007/s10107-007-0197-2.  Google Scholar

[12]

H. HeitschW. Römisch and C. Strugarek, Stability of multistage stochastic programs, SIAM J. Optim., 17 (2006), 511-525.  doi: 10.1137/050632865.  Google Scholar

[13]

T. Homem-de Mello and B. K. Pagnoncelli, Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective, European J. Oper. Res., 249 (2016), 188-199.  doi: 10.1016/j.ejor.2015.05.048.  Google Scholar

[14]

J. Jiang and Z. Chen, Quantitative stability of multistage stochastic programs via calm modifications, Oper. Res. Lett., 46 (2018), 543-547.  doi: 10.1016/j.orl.2018.08.007.  Google Scholar

[15]

R. Kovacevic and G. C. Pflug, Time consistency and information monotonicity of multiperiod acceptability functionals, in Advanced Financial Modelling, Radon Ser. Comput. Appl. Math., 8, Walter de Gruyter, Berlin, 2009,347–369. doi: 10.1515/9783110213140.347.  Google Scholar

[16]

C. Küchler, On stability of multistage stochastic programs, SIAM J. Optim., 19 (2008), 952-968.  doi: 10.1137/070690365.  Google Scholar

[17]

D. Kuhn, Generalized Bounds for Convex Multistage Stochastic Programs, Lecture Notes in Economics and Mathematical Systems, 548, Springer-Verlag, Berlin, 2006. doi: 10.1007/b138260.  Google Scholar

[18]

J. Liu and Z. Chen, Time consistent multi-period robust risk measures and portfolio selection models with regime-switching, European J. Oper. Res., 268 (2018), 373-385.  doi: 10.1016/j.ejor.2018.01.009.  Google Scholar

[19]

Y. LiuA. Pichler and H. Xu, Discrete approximation and quantification in distributionally robust optimization, Math. Oper. Res., 44 (2018), 19-37.  doi: 10.1287/moor.2017.0911.  Google Scholar

[20]

P. Mohajerin Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Math. Program., 171 (2018), 115-166.  doi: 10.1007/s10107-017-1172-1.  Google Scholar

[21]

T. Pennanen, Epi-convergent discretizations of multistage stochastic programs via integration quadratures, Math. Program., 116 (2009), 461-479.  doi: 10.1007/s10107-007-0113-9.  Google Scholar

[22]

G. C. Pflug and A. Pichler, Multistage Stochastic Optimization, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2014. doi: 10.1007/978-3-319-08843-3.  Google Scholar

[23]

G. C. Pflug and W. Römisch, Modeling, Measuring and Managing Risk, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812708724.  Google Scholar

[24]

A. B. Philpott and V. L. De Matos, Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion, European J. Oper. Res., 218 (2012), 470-483.  doi: 10.1016/j.ejor.2011.10.056.  Google Scholar

[25]

A. PhilpottV. de Matos and E. Finardi, On solving multistage stochastic programs with coherent risk measures, Oper. Res., 61 (2013), 957-970.  doi: 10.1287/opre.2013.1175.  Google Scholar

[26]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der mathematischen Wissenschaften, 317, Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[27]

W. Römisch, Stability of stochastic programming problems, in Stochastic Programming, Handbooks Oper. Res. Management Sci., 10, Elsevier Sci. B. V., Amsterdam, 2003,483–554. doi: 10.1016/S0927-0507(03)10008-4.  Google Scholar

[28]

A. Ruszczyński, Risk-averse dynamic programming for Markov decision processes, Math. Program., 125 (2010), 235-261.  doi: 10.1007/s10107-010-0393-3.  Google Scholar

[29]

A. Ruszczyński and A. Shapiro, Stochastic Programming, Elsevier, 2003. Google Scholar

[30]

A. Ruszczyński and A. Shapiro, Conditional risk mappings, Math. Oper. Res., 31 (2006), 544-561.  doi: 10.1287/moor.1060.0204.  Google Scholar

[31]

A. Shapiro, Inference of statistical bounds for multistage stochastic programming problems, Math. Methods Oper. Res., 58 (2003), 57-68.  doi: 10.1007/s001860300280.  Google Scholar

[32]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, MOS-SIAM Series on Optimization, SIAM, 2014. doi: 10.1137/1.9780898718751.  Google Scholar

[33]

T. Wang, A class of dynamic risk measures, Univ. British Columbia, 21 (1999). Google Scholar

[34]

C. Zhao and Y. Guan, Data-driven risk-averse two-stage stochastic program with $\zeta$-structure probability metrics, 2015. Available from: http://www.optimization-online.org/DB_FILE/2015/07/5014.pdf. Google Scholar

show all references

References:
[1]

S. A. Anthonisz, Asset pricing with partial-moments, J. Bank. Financ., 36 (2012), 2122-2135.  doi: 10.1016/j.jbankfin.2012.03.017.  Google Scholar

[2]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics, 28, Princeton University Press, Princeton, NJ, 2009. doi: 10.1515/9781400831050.  Google Scholar

[3]

A. J. Broganab and S. Stidham, Non-separation in the mean–lower-partial-moment portfolio optimization problem, European J. Oper. Res., 184 (2008), 701-710.  doi: 10.1016/j.ejor.2006.11.028.  Google Scholar

[4]

Z. Chen and J. Jiang, Stability analysis of optimization problems with $k$th order stochastic and distributionally robust dominance constraints induced by full random recourse, SIAM J. Optim., 28 (2018), 1396-1419.  doi: 10.1137/17M1120063.  Google Scholar

[5]

J. Dupačová and V. Kozmík, Structure of risk-averse multistage stochastic programs, OR Spectrum, 37 (2015), 559-582.  doi: 10.1007/s00291-014-0379-2.  Google Scholar

[6]

A. Eichhorn and W. Römisch, Polyhedral risk measures in stochastic programming, SIAM J. Optim., 16 (2005), 69-95.  doi: 10.1137/040605217.  Google Scholar

[7]

A. Eichhorn and W. Römisch, Stability of multistage stochastic programs incorporating polyhedral risk measures, Optimization, 57 (2008), 295-318.  doi: 10.1080/02331930701779930.  Google Scholar

[8]

P. C. Fishburn, Mean-risk analysis with risk associated with below-target returns, Amer. Econ. Rev., 67 (1977), 116-126.   Google Scholar

[9]

V. Guigues and W. Römisch, Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures, SIAM J. Optim., 22 (2012), 286-312.  doi: 10.1137/100811696.  Google Scholar

[10]

Y. Han and Z. Chen, Quantitative stability of full random two-stage stochastic programs with recourse, Optim. Lett., 9 (2015), 1075-1090.  doi: 10.1007/s11590-014-0827-6.  Google Scholar

[11]

H. Heitsch and W. Römisch, Scenario tree modeling for multistage stochastic programs, Math. Program., 118 (2009), 371-406.  doi: 10.1007/s10107-007-0197-2.  Google Scholar

[12]

H. HeitschW. Römisch and C. Strugarek, Stability of multistage stochastic programs, SIAM J. Optim., 17 (2006), 511-525.  doi: 10.1137/050632865.  Google Scholar

[13]

T. Homem-de Mello and B. K. Pagnoncelli, Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective, European J. Oper. Res., 249 (2016), 188-199.  doi: 10.1016/j.ejor.2015.05.048.  Google Scholar

[14]

J. Jiang and Z. Chen, Quantitative stability of multistage stochastic programs via calm modifications, Oper. Res. Lett., 46 (2018), 543-547.  doi: 10.1016/j.orl.2018.08.007.  Google Scholar

[15]

R. Kovacevic and G. C. Pflug, Time consistency and information monotonicity of multiperiod acceptability functionals, in Advanced Financial Modelling, Radon Ser. Comput. Appl. Math., 8, Walter de Gruyter, Berlin, 2009,347–369. doi: 10.1515/9783110213140.347.  Google Scholar

[16]

C. Küchler, On stability of multistage stochastic programs, SIAM J. Optim., 19 (2008), 952-968.  doi: 10.1137/070690365.  Google Scholar

[17]

D. Kuhn, Generalized Bounds for Convex Multistage Stochastic Programs, Lecture Notes in Economics and Mathematical Systems, 548, Springer-Verlag, Berlin, 2006. doi: 10.1007/b138260.  Google Scholar

[18]

J. Liu and Z. Chen, Time consistent multi-period robust risk measures and portfolio selection models with regime-switching, European J. Oper. Res., 268 (2018), 373-385.  doi: 10.1016/j.ejor.2018.01.009.  Google Scholar

[19]

Y. LiuA. Pichler and H. Xu, Discrete approximation and quantification in distributionally robust optimization, Math. Oper. Res., 44 (2018), 19-37.  doi: 10.1287/moor.2017.0911.  Google Scholar

[20]

P. Mohajerin Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Math. Program., 171 (2018), 115-166.  doi: 10.1007/s10107-017-1172-1.  Google Scholar

[21]

T. Pennanen, Epi-convergent discretizations of multistage stochastic programs via integration quadratures, Math. Program., 116 (2009), 461-479.  doi: 10.1007/s10107-007-0113-9.  Google Scholar

[22]

G. C. Pflug and A. Pichler, Multistage Stochastic Optimization, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2014. doi: 10.1007/978-3-319-08843-3.  Google Scholar

[23]

G. C. Pflug and W. Römisch, Modeling, Measuring and Managing Risk, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812708724.  Google Scholar

[24]

A. B. Philpott and V. L. De Matos, Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion, European J. Oper. Res., 218 (2012), 470-483.  doi: 10.1016/j.ejor.2011.10.056.  Google Scholar

[25]

A. PhilpottV. de Matos and E. Finardi, On solving multistage stochastic programs with coherent risk measures, Oper. Res., 61 (2013), 957-970.  doi: 10.1287/opre.2013.1175.  Google Scholar

[26]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der mathematischen Wissenschaften, 317, Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[27]

W. Römisch, Stability of stochastic programming problems, in Stochastic Programming, Handbooks Oper. Res. Management Sci., 10, Elsevier Sci. B. V., Amsterdam, 2003,483–554. doi: 10.1016/S0927-0507(03)10008-4.  Google Scholar

[28]

A. Ruszczyński, Risk-averse dynamic programming for Markov decision processes, Math. Program., 125 (2010), 235-261.  doi: 10.1007/s10107-010-0393-3.  Google Scholar

[29]

A. Ruszczyński and A. Shapiro, Stochastic Programming, Elsevier, 2003. Google Scholar

[30]

A. Ruszczyński and A. Shapiro, Conditional risk mappings, Math. Oper. Res., 31 (2006), 544-561.  doi: 10.1287/moor.1060.0204.  Google Scholar

[31]

A. Shapiro, Inference of statistical bounds for multistage stochastic programming problems, Math. Methods Oper. Res., 58 (2003), 57-68.  doi: 10.1007/s001860300280.  Google Scholar

[32]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, MOS-SIAM Series on Optimization, SIAM, 2014. doi: 10.1137/1.9780898718751.  Google Scholar

[33]

T. Wang, A class of dynamic risk measures, Univ. British Columbia, 21 (1999). Google Scholar

[34]

C. Zhao and Y. Guan, Data-driven risk-averse two-stage stochastic program with $\zeta$-structure probability metrics, 2015. Available from: http://www.optimization-online.org/DB_FILE/2015/07/5014.pdf. Google Scholar

Figure 1.  Scenario tree for Example 1
Figure 2.  Scenario trees of $ \mathit{\boldsymbol{\xi}} $ (left), $ f(\mathit{\boldsymbol{\xi}}) $ (central), $ \hat{\mathit{\boldsymbol{\xi}}} $ (right) for Example 2
[1]

Yuwei Shen, Jinxing Xie, Tingting Li. The risk-averse newsvendor game with competition on demand. Journal of Industrial & Management Optimization, 2016, 12 (3) : 931-947. doi: 10.3934/jimo.2016.12.931

[2]

Bin Zhou, Hailin Sun. Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049

[3]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019109

[4]

Xiulan Wang, Yanfei Lan, Wansheng Tang. An uncertain wage contract model for risk-averse worker under bilateral moral hazard. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1815-1840. doi: 10.3934/jimo.2017020

[5]

Kegui Chen, Xinyu Wang, Min Huang, Wai-Ki Ching. Compensation plan, pricing and production decisions with inventory-dependent salvage value, and asymmetric risk-averse sales agent. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1397-1422. doi: 10.3934/jimo.2018013

[6]

Bin Chen, Wenying Xie, Fuyou Huang, Juan He. Quality competition and coordination in a VMI supply chain with two risk-averse manufacturers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020100

[7]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[8]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019100

[9]

Bin Li, Jie Sun, Honglei Xu, Min Zhang. A class of two-stage distributionally robust games. Journal of Industrial & Management Optimization, 2019, 15 (1) : 387-400. doi: 10.3934/jimo.2018048

[10]

Yongchao Liu. Quantitative stability analysis of stochastic mathematical programs with vertical complementarity constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 451-460. doi: 10.3934/naco.2018028

[11]

Ke-Wei Ding, Nan-Jing Huang, Yi-Bin Xiao. Distributionally robust chance constrained problems under general moments information. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2923-2942. doi: 10.3934/jimo.2019087

[12]

Émilie Chouzenoux, Henri Gérard, Jean-Christophe Pesquet. General risk measures for robust machine learning. Foundations of Data Science, 2019, 1 (3) : 249-269. doi: 10.3934/fods.2019011

[13]

. Publisher Correction to: Probability, uncertainty and quantitative risk, volume 4. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 7-. doi: 10.1186/s41546-019-0041-7

[14]

Jie Sun, Honglei Xu, Min Zhang. A new interpretation of the progressive hedging algorithm for multistage stochastic minimization problems. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1655-1662. doi: 10.3934/jimo.2019022

[15]

Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2020, 16 (4) : 2029-2044. doi: 10.3934/jimo.2019041

[16]

Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control & Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165

[17]

Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6835-6853. doi: 10.3934/dcds.2016097

[18]

Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357

[19]

Alexey G. Mazko. Positivity, robust stability and comparison of dynamic systems. Conference Publications, 2011, 2011 (Special) : 1042-1051. doi: 10.3934/proc.2011.2011.1042

[20]

Harald Held, Gabriela Martinez, Philipp Emanuel Stelzig. Stochastic programming approach for energy management in electric microgrids. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 241-267. doi: 10.3934/naco.2014.4.241

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (27)
  • HTML views (213)
  • Cited by (0)

Other articles
by authors

[Back to Top]