• Previous Article
    Decision framework for location and selection of container multimodal hubs: A case in china under the belt and road initiative
  • JIMO Home
  • This Issue
  • Next Article
    Preserving relational contract stability of fresh agricultural product supply chains
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]

Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021024

[2]

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

[3]

Jianxun Liu, Shengjie Li, Yingrang Xu. Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021083

[4]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

[5]

Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025

[6]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[7]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022

[8]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040

[9]

Yves Capdeboscq, Shaun Chen Yang Ong. Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3857-3887. doi: 10.3934/dcdsb.2020228

[10]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409

[11]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[12]

Mohammed Abdelghany, Amr B. Eltawil, Zakaria Yahia, Kazuhide Nakata. A hybrid variable neighbourhood search and dynamic programming approach for the nurse rostering problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2051-2072. doi: 10.3934/jimo.2020058

[13]

Kai Kang, Taotao Lu, Jing Zhang. Financing strategy selection and coordination considering risk aversion in a capital-constrained supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021042

[14]

Vladimir Gaitsgory, Ilya Shvartsman. Linear programming estimates for Cesàro and Abel limits of optimal values in optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021102

[15]

Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020179

[16]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015

[17]

Qing Liu, Bingo Wing-Kuen Ling, Qingyun Dai, Qing Miao, Caixia Liu. Optimal maximally decimated M-channel mirrored paraunitary linear phase FIR filter bank design via norm relaxed sequential quadratic programming. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1993-2011. doi: 10.3934/jimo.2020055

[18]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[19]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[20]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (62)
  • HTML views (367)
  • Cited by (0)

Other articles
by authors

[Back to Top]