September  2021, 17(5): 2415-2440. 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  September 2021 Early access  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 and Management Optimization, 2021, 17 (5) : 2415-2440. doi: 10.3934/jimo.2020075
References:
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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.

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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.

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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.

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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.

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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.

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

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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.

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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.

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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.

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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.

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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.

[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.

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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.

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

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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.

[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.

[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.

[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.

[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.

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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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[30]

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

[31]

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

[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.

[33]

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

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[30]

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

[31]

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

[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.

[33]

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

[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.

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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