Abstract

Distributionally robust optimization involves various probability measures in its problem formulation. They can be bundled to constitute a risk functional. For this equivalence, risk functionals constitute a fundamental building block in distributionally robust stochastic programming. Multistage programming requires conditional versions of risk functionals to reassess future risk after partial realizations and after preceding decisions. This paper discusses a construction of the conditional counterpart of a risk functional by passing its genuine characteristics to its conditional counterparts. It turns out that the conditional risk functionals could be different from the nested analogues of the original (law invariant) risk measure. We also discuss an implication to formulations of distributionally robust stochastic programming and a relation to stochastic games.

Keywords

  1. multistage stochastic programming
  2. distributional robustness
  3. conditional risk measures
  4. dynamic equations
  5. stochastic games
  6. rectangularity

MSC codes

  1. 90C15
  2. 60B05
  3. 62P05
  4. 90C31
  5. 90C08

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References

1.
L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Birkhäuser Verlag, Basel, 2005, https://doi.org/10.1007/978-3-7643-8722-8.
2.
P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, Coherent measures of risk, Math. Finance, 9 (1999), pp. 203--228.
3.
P. M. Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Math. Program., 171 (2016), pp. 115--166, https://doi.org/10.1007/s10107-017-1172-1.
4.
H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 2nd ed., De Gruyter Stud. Math. 27, De Gruyter, Berlin, 2004.
5.
M. Frittelli and E. Rosazza Gianin, Dynamic convex risk measures, in Risk Measures for the 21st Century, G. Szegö, ed., John Wiley & Sons, Chichester, UK, 2005, pp. 227--248.
6.
G. A. Hanasusanto, V. Roitch, D. Kuhn, and W. Wiesemann, A distributionally robust perspective on uncertainty quantification and chance constrained programming, Math. Program., 151 (2015), pp. 35--62, https://doi.org/10.1007/s10107-015-0896-z.
7.
A. Jaśkiewicz and A. S. Nowak, Zero-sum stochastic games, in Handbook of Dynamic Game Theory, T. Basar and G. Zaccour, eds., Springer, Cham, 2018, pp. 215--279.
8.
O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002, https://doi.org/10.1007/b98838.
9.
G. Ch. Pflug and A. Pichler, Multistage Stochastic Optimization, Springer Ser. Oper. Res. Financ. Eng., Springer, Cham, 2014, https://doi.org/10.1007/978-3-319-08843-3.
10.
G. Ch. Pflug and A. Pichler, Time-inconsistent multistage stochastic programs: Martingale bounds, European J. Oper. Res., 249 (2016), pp. 155--163, https://doi.org/10.1016/j.ejor.2015.02.033.
11.
G. Ch. Pflug and A. Pichler, Time-consistent decisions and temporal decomposition of coherent risk functionals, Math. Oper. Res., 41 (2016), pp. 682--699, https://doi.org/10.1287/moor.2015.0747.
12.
F. Riedel, Dynamic coherent risk measures, Stochastic Process. Appl., 112 (2004), pp. 185--200, https://doi.org/10.1016/j.spa.2004.03.004.
13.
R. T. Rockafellar and S. Uryasev, Optimization of Conditional Value-at-Risk, J. Risk, 2 (2000), pp. 21--41, https://doi.org/10.21314/JOR.2000.038.
14.
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998, https://doi.org/10.1007/978-3-642-02431-3.
15.
A. Ruszczyński and A. Shapiro, Conditional risk mappings, Math. Oper. Res., 31 (2006), pp. 544--561, https://doi.org/10.1287/moor.1060.0204.
16.
A. Ruszczyński and A. Shapiro, Optimization of convex risk functions, Math. Oper. Res., 31 (2006), pp. 433--452.
17.
A. Shapiro, Rectangular sets of probability measures, Oper. Res., 64 (2016), pp. 528--541, https://doi.org/10.1287/opre.2015.1466.
18.
A. Shapiro, Interchangeability principle and dynamic equations in risk averse stochastic programming, Oper. Res. Lett., 45 (2017), pp. 377--381, https://doi.org/10.1016/j.orl.2017.05.008.
19.
A. Shapiro, D. Dentcheva, and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, 2nd ed., MOS-SIAM Ser. Optim. 16, SIAM, Philadelphia, 2014, https://doi.org/10.1137/1.9781611973433.
20.
L. S. Shapley, Stochastic games, Proc. Natl. Acad. Sci. USA, 39 (1953), pp. 1095--1100.
21.
M. Sion, On general minimax theorems., Pacific J. Math., 8 (1958), pp. 171--176, https://doi.org/pjm/1103040253.
22.
C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, AMS, Providence, RI, 2003, https://doi.org/10.1090/gsm/058.
23.
W. Wiesemann, D. Kuhn, and M. Sim, Distributionally robust convex optimization, Oper. Res., 62 (2014), pp. 1358--1376, https://doi.org/10.1287/opre.2014.1314.
24.
J. Zhen, D. Kuhn, and W. Wiesemann, Mathematical Foundations of Robust and Distributionally Robust Optimization, preprint, https://arxiv.org/abs/2105.00760, 2021.

Information & Authors

Information

Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 3044 - 3067
ISSN (online): 1095-7189

History

Submitted: 7 January 2021
Accepted: 5 September 2021
Published online: 30 November 2021

Keywords

  1. multistage stochastic programming
  2. distributional robustness
  3. conditional risk measures
  4. dynamic equations
  5. stochastic games
  6. rectangularity

MSC codes

  1. 90C15
  2. 60B05
  3. 62P05
  4. 90C31
  5. 90C08

Authors

Affiliations

Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 416228727 - SFB 1410

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