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Exact penalization, level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints

Exact penalization, level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints
Exact penalization, level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints
Level function methods and cutting-plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance (SSD) constraints. A level function method requires an exact penalization setup because it can only be applied to the objective function, not the constraints. Slater constraint qualification (SCQ) is often needed for deriving exact penalization. It is well known that SSD usually does not satisfy SCQ and various relaxation schemes have been proposed so that the relaxed problem satisfies the SCQ. In this paper, we show that under some moderate conditions the desired constraint qualification can be guaranteed through some appropriate reformulation of the constraints rather than relaxation. Exact penalization schemes based on $L_1$-norm and $L_\infty$-norm are subsequently derived through Robinson's error bound on convex systems and Clarke's exact penalty function theorem. Moreover, we propose a modified cutting-plane method which constructs a cutting-plane through the maximum of the reformulated constraint functions. In comparison with the existing cutting-plane methods, it is numerically more efficient because only a single cutting plane is constructed and added at each iteration. We have carried out a number of numerical experiments and the results show that our methods display better performances particularly in the case when the underlying functions are nonlinear w.r.t. decision variables.


Read More: http://epubs.siam.org/doi/abs/10.1137/110850815
1052-6234
602-631
Sun, Hailin
eee2e5fb-018b-45d4-b599-08209509663c
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Meskarian, Rudabeh
932d1dac-784b-4f24-bdda-5ea34a16d8a2
Wang, Yong
a7913cc9-1250-4f71-ab1f-93480043bb6a
Sun, Hailin
eee2e5fb-018b-45d4-b599-08209509663c
Xu, Huifu
d3200e0b-ad1d-4cf7-81aa-48f07fb1f8f5
Meskarian, Rudabeh
932d1dac-784b-4f24-bdda-5ea34a16d8a2
Wang, Yong
a7913cc9-1250-4f71-ab1f-93480043bb6a

Sun, Hailin, Xu, Huifu, Meskarian, Rudabeh and Wang, Yong (2013) Exact penalization, level function method and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints. SIAM Journal on Optimization, 23 (1), 602-631. (doi:10.1137/110850815).

Record type: Article

Abstract

Level function methods and cutting-plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance (SSD) constraints. A level function method requires an exact penalization setup because it can only be applied to the objective function, not the constraints. Slater constraint qualification (SCQ) is often needed for deriving exact penalization. It is well known that SSD usually does not satisfy SCQ and various relaxation schemes have been proposed so that the relaxed problem satisfies the SCQ. In this paper, we show that under some moderate conditions the desired constraint qualification can be guaranteed through some appropriate reformulation of the constraints rather than relaxation. Exact penalization schemes based on $L_1$-norm and $L_\infty$-norm are subsequently derived through Robinson's error bound on convex systems and Clarke's exact penalty function theorem. Moreover, we propose a modified cutting-plane method which constructs a cutting-plane through the maximum of the reformulated constraint functions. In comparison with the existing cutting-plane methods, it is numerically more efficient because only a single cutting plane is constructed and added at each iteration. We have carried out a number of numerical experiments and the results show that our methods display better performances particularly in the case when the underlying functions are nonlinear w.r.t. decision variables.


Read More: http://epubs.siam.org/doi/abs/10.1137/110850815

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Published date: 2013
Organisations: Operational Research

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Local EPrints ID: 390730
URI: http://eprints.soton.ac.uk/id/eprint/390730
ISSN: 1052-6234
PURE UUID: d3b255cc-69ee-4a43-a308-cf4f0a10f7cd
ORCID for Huifu Xu: ORCID iD orcid.org/0000-0001-8307-2920

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Date deposited: 06 Apr 2016 14:43
Last modified: 22 Oct 2019 00:46

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