More Web Proxy on the site http://driver.im/

article

Stochastic Optimization Problems with CVaR Risk Measure and Their Sample Average Approximation

Authors:

M. GohAuthors Info & Claims

Journal of Optimization Theory and Applications, Volume 146, Issue 2

Pages 399 - 418

https://doi.org/10.1007/s10957-010-9676-3

Published: 01 August 2010 Publication History

Abstract

We provide a refined convergence analysis for the SAA (sample average approximation) method applied to stochastic optimization problems with either single or mixed CVaR (conditional value-at-risk) measures. Under certain regularity conditions, it is shown that 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. In addition, it is shown that, as the sample size increases, the difference of the optimal values between the SAA problems and the original problem tends to zero with probability approaching one exponentially fast.

References

[1]

Anderson, F., Mausser, H., Rosen, D., Uryasev, S.: Credit risk optimization with conditional value-at-risk criterion. Math. Program. 89, 273---291 (2001)

[2]

Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203---228 (1999)

[3]

Hurlimann, W.: Conditional value-at-risk bounds for compound Poisson risks and a normal approximation. J. Appl. Math. 3, 141---153 (2003)

[4]

Krokhmal, P., Palmquist, J., Uryasev, S.: Portfolio optimization with conditional value-at-risk objective and constraints. J. Risk 4, 43---68 (2002)

[5]

Lüthi, H.-J., Doege, J.: Convex risk measures for portfolio optimization and concepts of flexibility. Math. Program. 104, 541---559 (2005)

Digital Library

[6]

Pang, J.-S., Leyffer, S.: On the global minimization of the value-at-risk. Optim. Methods Softw. 19, 611---631 (2004)

[7]

Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443---1471 (2002)

[8]

Acerbi, C.: Spectral measures of risk: A coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505---1518 (2002)

[9]

Bogentoft, E., Romeijn, H.E., Uryasev, S.: Asset/liability management for pension funds using CVaR constraints. J. Risk Finance 3, 57---71 (2001)

[10]

Chen, W., Sim, M., Sun, J., Teo, C.P.: From CVaR to uncertainty set: Implications in joint chance constrained optimization. Oper. Res. (2009).

Digital Library

[11]

Küzi-Bay, A., Mayer, J.: Computational aspects of minimizing conditional value-at-risk. Comput. Manag. Sci. 3, 3---27 (2006)

[12]

Shapiro, A.: Stochastic programming by Monte Carlo simulation methods. Published electronically in: Stoch. Program. E-Print Ser. (2000)

[13]

Santoso, T., Ahmed, S., Goetschalckx, M., Shapiro, A.: A stochastic programming approach for supply chain network design under uncertainty. Eur. J. Oper. Res. 67, 96---115 (2005)

[14]

Meng, F., Xu, H.: A regularized sample average approximation method for stochastic mathematical programs with nonsmooth equality constraints. SIAM J. Optim. 17, 891---919 (2006)

Digital Library

[15]

Shapiro, A.: Stochastic mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 128, 223---243 (2006)

Digital Library

[16]

Xu, H., Meng, F.: Convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints. Math. Oper. Res. 32, 648---668 (2007)

Digital Library

[17]

Artstein, Z., Vitale, R.A.: A strong law of large numbers for random compact sets. Ann. Probab. 3, 879---882 (1975)

[18]

Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

[19]

Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

[20]

Wets, R.: Stochastic programming. In: Nemhauser, G.L., Rinnooy Kan, A.H.H., Todd, M.J. (eds.) Handbooks in OR & MS, vol. 10, pp. 573---629. North-Holland, Amsterdam (1989)

Digital Library

[21]

Homen-De-Mello, T.: Estimation of derivatives of nonsmooth performance measures in regenerative systems. Math. Oper. Res. 26, 741---768 (2001)

Digital Library

[22]

Ruszczyński, A., Shapiro, A.: Optimality and duality in stochastic programming. In: Ruszczyński, A., Shapiro, A. (eds.) Handbooks in OR & MS Stochastic Programming, vol. 10, pp. 65---140. North-Holland, Amsterdam (2003)

[23]

Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation. Optimization 57, 395---418 (2008)

[24]

Qi, L., Shapiro, A., Ling, C.: Differentiability and semismoothness properties of integral functions and their applications. Math. Program. 102, 223---248 (2005)

Digital Library

Cited By

Yiyong Feng Palomar DRubio F(2015)Robust Optimization of Order ExecutionIEEE Transactions on Signal Processing10.1109/TSP.2014.238628863:4(907-920)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1109/TSP.2014.2386288
Sun HXu HWang Y(2014)Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex FunctionsJournal of Optimization Theory and Applications10.1007/s10957-012-0127-1161:1(257-284)Online publication date: 1-Apr-2014
https://dl.acm.org/doi/10.1007/s10957-012-0127-1

Stochastic Optimization Problems with CVaR Risk Measure and Their Sample Average Approximation

Recommendations

A smoothing sample average approximation method for stochastic optimization problems with CVaR risk measure

This paper is concerned with solving single CVaR and mixed CVaR minimization problems. A CHKS-type smoothing sample average approximation (SAA) method is proposed for solving these two problems, which retains the convexity and smoothness of the original ...
Sample Average Approximation Method for Compound Stochastic Optimization Problems

The paper studies stochastic optimization (programming) problems with compound functions containing expectations and extreme values of other random functions as arguments. Compound functions arise in various applications. A typical example is a variance ...
Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach

This paper presents an asymptotic analysis of a Monte Carlo method, variously known as sample average approximation (SAA) or sample path optimization (SPO), for a general two-stage stochastic minimization problem. We study the case when the second-stage ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Optimization Theory and Applications

Journal of Optimization Theory and Applications Volume 146, Issue 2

August 2010

297 pages

ISSN:0022-3239

Issue’s Table of Contents

Copyright © Copyright © 2010 Springer Science+Business Media, LLC.

Publisher

Plenum Press

United States

Publication History

Published: 01 August 2010

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 20 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Yiyong Feng Palomar DRubio F(2015)Robust Optimization of Order ExecutionIEEE Transactions on Signal Processing10.1109/TSP.2014.238628863:4(907-920)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1109/TSP.2014.2386288
Sun HXu HWang Y(2014)Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex FunctionsJournal of Optimization Theory and Applications10.1007/s10957-012-0127-1161:1(257-284)Online publication date: 1-Apr-2014
https://dl.acm.org/doi/10.1007/s10957-012-0127-1

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents