Consistency of Sample Estimates of Risk Averse Stochastic Programs

Shapiro, Alexander

doi:10.1017/s0021900200013541

Cited by 11 publications

(10 citation statements)

References 7 publications

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“…It is possible to show that, under mild regularity conditions, the optimal value and optimal solutions of the SAA problem converge w.p.1 to their true counterparts as the sample size N tends to infinity (cf. [23]).…”

Section: Consider the Uncertainty Setmentioning

confidence: 99%

Distributionally Robust Stochastic Programming

Shapiro¹

2017

SIAM J. Optim.

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129

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In this paper we study distributionally robust stochastic programming in a setting where there is a specified reference probability measure and the uncertainty set of probability measures consists of measures in some sense close to the reference measure. We discuss law invariance of the associated worst case functional and consider two basic constructions of such uncertainty sets. Finally we illustrate some implications of the property of law invariance.

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Section: Consider the Uncertainty Setmentioning

confidence: 99%

Distributionally Robust Stochastic Programming

Shapiro¹

2017

SIAM J. Optim.

Self Cite

129

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“…We want to show 0 ∩ * = (35) for any Q ∈ . By definition, for any ∈ , there exist s ∈ 0 and c ∈ such that = s + c .…”

Section: Moment Problemsmentioning

confidence: 99%

Convergence Analysis for Distributionally Robust Optimization and Equilibrium Problems

Sun

2016

Mathematics of OR

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In this paper, we study distributionally robust optimization approaches for a one-stage stochastic minimization problem, where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions, which contains the true distribution and optimal decision is taken on the basis of the worst-possible distribution from that set. We consider the case when the distributional set (which is also known as the ambiguity set) varies and its impact on the optimal value and the optimal solutions. A typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size. The analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming. The discussion is extended briefly to a stochastic Nash equilibrium problem where each player takes a robust action on the basis of the worst subjective expected objective values.

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“…According to the definition of the time consistency property of risk averse measures given in Homem-de-Mello and Pagnoncelli (2016), it is not difficult to prove that ECSD in tree n is a strategic node-based time-consistent functional. See also Kormik and Morton (2015), Pflug and Pichler (2015), Rudloff et al (2014), Ruszczyński (2010), Shapiro (2009), Shapiro and Pichler (2016), among others. 4.…”

Section: Strategic Node-based Time-consistent Sd Measure For Operational Functionsmentioning

confidence: 99%

On capacity expansion planning under strategic and operational uncertainties based on stochastic dominance risk averse management

Escudero

Monge

2018

Comput Manag Sci

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A new scheme for dealing with uncertainty in scenario trees is presented for dynamic mixed 0-1 optimization problems with strategic and operational stochastic parameters. Let us generically name this type of problems as capacity expansion planning (CEP) in a given system, e.g., supply chain, production, rapid transit network, energy generation and transmission network, etc. The strategic scenario tree is usually a multistage one, and the replicas of the strategic nodes root structures in the form of either a special scenario graph or a two-stage scenario tree, depending on the type of operational activity in the system. Those operational scenario structures impact in the constraints of the model and, thus, in the decomposition methodology for solving usually large-scale problems. This work presents the modeling framework for some of the risk neutral and risk averse measures to consider for CEP problem solving. Two types of risk averse measures are considered. The first one is a time-inconsistent mixture of the chance-constrained and second-order stochastic dominance (SSD) functionals of the value of a given set of functions up to the strategic nodes in selected stages along the time horizon, The second type is a strategic node-based time-consistent SSD functional for the set of operational scenarios in the strategic nodes at selected stages. A specialization of the nested stochastic decomposition methodology for that problem solving is outlined. Its advantages and drawbacks as well as the framework for some schemes to, at least, partially avoid those drawbacks are also presented.

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Consistency of Sample Estimates of Risk Averse Stochastic Programs

Cited by 11 publications

References 7 publications

Distributionally Robust Stochastic Programming

Distributionally Robust Stochastic Programming

Convergence Analysis for Distributionally Robust Optimization and Equilibrium Problems

On capacity expansion planning under strategic and operational uncertainties based on stochastic dominance risk averse management

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