Conditional Distributionally Robust Functionals
Abstract
Many decisions, in particular decisions in a managerial context, are subject to uncertainty. Risk measures cope with uncertainty by involving more than one candidate probability. The corresponding risk averse decision takes all potential candidate probabilities into account and is robust with respect to all potential probabilities. This paper considers conditional robust decision making, where decisions are subject to additional prior knowledge or information. The literature discusses various definitions to characterize the corresponding conditional risk measure, which determines further the decision. The aim of this paper is to compare two different approaches for the construction of conditional functionals used in multistage distributionally robust optimization. As an application, we discuss conditional counterparts of a distance between probability measures.
Funding: A. Shapiro was partly supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0244]. A. Pichler was funded by the Deutsche Forschungsgemeinschaft [Project 416228727–SFB 1410].
References
[1]
Bayraksan G, Love DK (2015) Data-driven stochastic programming using phi-divergences. Tutorials in Operations Research (INFORMS, Catonsville, MD), 1563–1581.
[2]
Ben-Tal A, Teboulle M (2007) An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance 17:449–476.
[3]
Csiszár I (1963) Eine informationstheoretische ungleichung und ihre anwendung auf den beweis der ergodizität von markoffschen ketten. Magyar. Tud. Akad. Mat. Kutato Int. Kozls. 8:85–105.
[4]
Dellacherie C, Meyer P-A (1988) Probabilities and Potential (North-Holland Publishing, Amsterdam).
[5]
Ellsberg D (1961) Risk, ambiguity, and the savage axioms. Quart. J. Econom. 75(4):643–669.
[6]
Esteban-Pérez A, Morales JM (2021) Distributionally robust stochastic programs with side information based on trimmings. Math. Programming 195:1069–1105.
[7]
Föllmer H, Schied A (2004) Stochastic Finance: An Introduction in Discrete Time (De Gruyter, Boston).
[8]
Hanasusanto GA, Roitch V, Kuhn D, Wiesemann W (2015) A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math. Programming 151(1):35–62.
[9]
Keith AJ, Ahner DK (2019) A survey of decision making and optimization under uncertainty. Ann. Oper. Res. 300(2):319–353.
[10]
Knight FH (1921) Risk, Uncertainty and Profit (Library of Economics and Liberty, Carmel, IN).
[11]
Liu Y, Pichler A, Xu H (2018) Discrete approximation and quantification in distributionally robust optimization. Math. Oper. Res. 44(1):19–37.
[12]
Morimoto T (1963) Markov processes and the H-theorem. J. Phys. Soc. Japan 18(3):328–331.
[13]
Pflug GCh (2009) Version-independence and nested distributions in multistage stochastic optimization. SIAM J. Optim. 20:1406–1420.
[14]
Pflug GCh, Pichler A (2012) A distance for multistage stochastic optimization models. SIAM J. Optim. 22(1):1–23.
[15]
Pflug GCh, Pichler A (2015) Time-inconsistent multistage stochastic programs: Martingale bounds. Eur. J. Oper. Res. 249(1):155–163.
[16]
Pichler A, Shapiro A (2021) Mathematical foundations of distributionally robust multistage optimization. SIAM J. Optim. 31(4):3044–3067.
[17]
Rahimian H, Mehrotra S (2019) Distributionally robust optimization: A review. Preprint, submitted August 13, https://doi.org/10.48550/arXiv.1908.05659.
[18]
Riedel F (2004) Dynamic coherent risk measures. Stochastic Process. Appl. 112(2):185–200.
[19]
Ruszczyński A, Shapiro A (2006) Conditional risk mappings. Math. Oper. Res. 31(3):544–561.
[20]
Scarf H (1958) A min-max solution of an inventory problem. Studies in the Mathematical Theory of Inventory and Production (Stanford University Press, Stanford, CA), 201–209.
[21]
Shapiro A (2012) Minimax and risk averse multistage stochastic programming. Eur. J. Oper. Res. 219(3):719–726.
[22]
Shapiro A (2017) Interchangeability principle and dynamic equations in risk averse stochastic programming. Oper. Res. Lett. 45(4):377–381.
[23]
Shapiro A, Dentcheva D, Ruszczyński A (2021) Lectures on Stochastic Programming. MOS-SIAM Series on Optimization, 3rd ed. (SIAM, Philadelphia).
[24]
Sun H, Xu H (2016) Convergence analysis for distributionally robust optimization and equilibrium problems. Math. Oper. Res. 41(2):377–401.
[25]
Villani C (2003) Topics in Optimal Transportation, vol. 58 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI).
[26]
Wiesemann W, Kuhn D, Sim M (2014) Distributionally robust convex optimization. Oper. Res. 62(6):1358–1376.
Index Terms
- Conditional Distributionally Robust Functionals
Index terms have been assigned to the content through auto-classification.
Recommendations
Mathematical Foundations of Distributionally Robust Multistage Optimization
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 ...
Technical Note—Time Inconsistency of Optimal Policies of Distributionally Robust Inventory Models
The authors extend previous studies of time inconsistency to risk averse (distributionally robust) inventory models and show that time inconsistency is not unique to robust multistage decision making, but may happen for a large class of risk averse/...
In this paper, we investigate optimal policies of distributionally robust (risk averse) inventory models. We demonstrate that if the respective risk measures are not strictly monotone, then there may exist infinitely many optimal policies that are not ...
Distributionally Robust Markov Decision Processes
We consider Markov decision processes where the values of the parameters are uncertain. This uncertainty is described by a sequence of nested sets (that is, each set contains the previous one), each of which corresponds to a probabilistic guarantee for ...
Comments
Information & Contributors
Information
Published In
Copyright © 2023 The Author(s).
This work is licensed under a Creative Commons Attribution 4.0 International License. You are free to copy, distribute, transmit and adapt this work, but you must attribute this work as “Operations Research. Copyright © 2023 The Author(s). https://doi.org/10.1287/opre.2023.2470, used under a Creative Commons Attribution License: https://creativecommons.org/licenses/by/4.0/.”
Publisher
INFORMS
Linthicum, MD, United States
Publication History
Published: 29 June 2023
Accepted: 16 March 2023
Received: 21 May 2022
Author Tag
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 12 Jan 2025