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]

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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

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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

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

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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

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

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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

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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

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

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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

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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

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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
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