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Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications
Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications
Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem.
smoothing method, sample average approximation, stationary points, error bound
0025-5610
371-401
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Zhang, Dali
f0f07f05-a0ee-4a3a-98c8-b24d73ce1a59
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Zhang, Dali
f0f07f05-a0ee-4a3a-98c8-b24d73ce1a59

Xu, Huifu and Zhang, Dali (2009) Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications. Mathematical Programming, 119 (2), 371-401. (doi:10.1007/s10107-008-0214-0).

Record type: Article

Abstract

Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem.

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More information

Published date: July 2009
Keywords: smoothing method, sample average approximation, stationary points, error bound
Organisations: Operational Research

Identifiers

Local EPrints ID: 79531
URI: http://eprints.soton.ac.uk/id/eprint/79531
ISSN: 0025-5610
PURE UUID: 015c07c4-4b0b-4063-a2d5-c0dd1bc443cc
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

Catalogue record

Date deposited: 16 Mar 2010
Last modified: 08 Jan 2022 02:57

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Contributors

Author: Huifu Xu ORCID iD
Author: Dali Zhang

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