Stochastic optimization problems with CVaR risk measure and their sample average approximation
type: Item
instance of: scholarly article
title: Stochastic optimization problems with CVaR risk measure and their sample average approximation
zbMATH Open document ID: 1213.91161
author: Fanwen Meng
author: Jie Sun
author: Mark Goh
published in: Journal of Optimization Theory and Applications
publication date: 2010-11-10T00:00:00Z
review text: Considered are the single CVaR and mixed CVaR minimization problems \[ \min_{x\in \mathcal{X}}\mathrm{CVaR}_{\alpha }(x) \eqno(1) \] and \[ \min_{x\in \mathcal{X}}\left\{ \lambda _{1}\mathrm{CVaR}_{\alpha _{1}}(x) +\cdots +\lambda _{J}\mathrm{CVaR}_{\alpha _{J}}(x) \right\} \eqno(2) \] where \(x\in \mathcal{X}\subseteq R^{n}\) is a control vector, \(\mathcal{X}\) denotes the set of feasible decisions, \(\alpha _{i}\in (0,1)\) denote the probability levels and \(\lambda _{i}>0\) represents weights with \(\sum_{i=1}^{J}\lambda _{i}=1,i=1,\dots ,J\). The aim of this paper is the analysis of a Monte Carlo simulation-based sampling approach to problems (1) and (2), called the sample average approximation (SAA) method. The authors show that under certain regularity conditions any accumulation point of the weak GKKT (generalized Karush-Kuhn-Tucker) points produced by the SAA method is almost surely a weak stationary point of the original CVaR or mixed CVaR optimization problems. They show that, as the sample size increases the difference of the optimal value between the SAA problems and the original problem tends to zero with probability approaching to one exponentially fast.
Mathematics Subject Classification ID: 91G60
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 90C15
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 90C59
zbMATH DE Number: 5814309
zbMATH Keywords: stochastic optimization
zbMATH Keywords: sample average approximation
reviewed by: I. M. Stancu-Minasian
MaRDI profile type: Publication
full work available at URL: s10957-010-9676-3
OpenAlex ID: W2037656742
cites work: Credit risk optimization with conditional Value-at-Risk criterion
cites work: Coherent measures of risk
cites work: Conditional value-at-risk bounds for compound Poisson risks and a normal approximation
cites work: Convex risk measures for portfolio optimization and concepts of flexibility
cites work: On the global minimization of the value-at-risk
cites work: From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization
cites work: Computational aspects of minimizing conditional value-at-risk
cites work: A stochastic programming approach for supply chain network design under uncertainty
cites work: A Regularized Sample Average Approximation Method for Stochastic Mathematical Programs with Nonsmooth Equality Constraints
cites work: Stochastic programming with equilibrium constraints
cites work: Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints
cites work: A strong law of large numbers for random compact sets
cites work: Optimization and nonsmooth analysis
cites work: Convex Analysis
cites work: Estimation of Derivatives of Nonsmooth Performance Measures in Regenerative Systems
cites work: Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation
cites work: Differentiability and semismoothness properties of integral functions and their applications
DOI: 10.1007/S10957-010-9676-3
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description: scientific article; zbMATH DE number 5814309