A Linear Decision-Based Approximation Approach to Stochastic Programming
Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of ad...
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| Vydáno v: | Operations research Ročník 56; číslo 2; s. 344 - 357 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Linthicum, MD
INFORMS
01.03.2008
Institute for Operations Research and the Management Sciences |
| Témata: | |
| ISSN: | 0030-364X, 1526-5463 |
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| Abstract | Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse. |
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| AbstractList | Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse. Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse. [PUBLICATION ABSTRACT] Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the so-called second-order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a two-stage stochastic optimization problem with complete recourse. Subject classifications: programming: stochastic. Area of review: Stochastic Models. |
| Audience | Trade |
| Author | Zhang, Jiawei Sim, Melvyn Chen, Xin Sun, Peng |
| Author_xml | – sequence: 1 fullname: Chen, Xin – sequence: 2 fullname: Sim, Melvyn – sequence: 3 fullname: Sun, Peng – sequence: 4 fullname: Zhang, Jiawei |
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| References_xml | – ident: B12 – ident: B9 – ident: B14 – ident: B10 – ident: B3 – ident: B20 – ident: B1 – ident: B27 – ident: B7 – ident: B5 – ident: B29 – ident: B25 – ident: B23 – ident: B21 – ident: B18 – ident: B16 – ident: B31 – ident: B8 – ident: B11 – ident: B13 – ident: B2 – ident: B26 – ident: B4 – ident: B28 – ident: B6 – ident: B24 – ident: B22 – ident: B17 – ident: B32 – ident: B15 – ident: B30 – ident: B19 – ident: B6 doi: 10.1287/msom.1050.0081 – start-page: 201 volume-title: Studies in the Mathematical Theory of Inventory and Production year: 1958 ident: B30 – ident: B19 doi: 10.1287/mnsc.1.3-4.197 – ident: B22 doi: 10.1137/S0895479896298130 – ident: B11 doi: 10.1007/s10107-005-0677-1 – volume: 17 start-page: 173 year: 1955 ident: B2 publication-title: J. Roy. Statist. Soc. Ser. B doi: 10.1111/j.2517-6161.1955.tb00191.x – ident: B7 doi: 10.1007/s10107-003-0454-y – ident: B16 doi: 10.1287/mnsc.6.1.73 – ident: B4 doi: 10.1016/S0167-6377(99)00016-4 – volume-title: Introduction to Stochastic Programming year: 1997 ident: B12 – ident: B31 doi: 10.1007/0-387-26771-9_4 – ident: B25 doi: 10.1007/s00186-006-0104-2 – ident: B21 doi: 10.1007/s10107-005-0597-0 – ident: B26 doi: 10.1007/s10107-003-0425-3 – volume-title: Stochastic Programming, Handbooks in Operations Research and Management Science year: 2003 ident: B29 – ident: B1 doi: 10.1287/opre.1070.0428 – ident: B3 doi: 10.1287/moor.23.4.769 – ident: B8 doi: 10.1007/s10107-003-0396-4 – ident: B17 doi: 10.1287/opre.1070.0441 – ident: B27 doi: 10.1080/02331930701421046 – ident: B9 doi: 10.1287/opre.1030.0065 – ident: B13 doi: 10.1007/s10107-003-0499-y – ident: B20 doi: 10.1287/moor.1040.0094 – ident: B32 doi: 10.1287/opre.21.5.1154 – ident: B15 doi: 10.1109/TAC.2006.875041 – ident: B28 doi: 10.1137/S1052623403430099 – ident: B24 doi: 10.1007/s10107-005-0678-0 – ident: B23 doi: 10.1137/S1052623496305717 – ident: B5 doi: 10.1007/PL00011380 |
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| Snippet | Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of... |
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| SubjectTerms | Analysis Applied sciences Approximation Covariance Decision making Decision theory. Utility theory Exact sciences and technology Linear programming Mathematical vectors Mathematics Matrices Objective functions Operational research and scientific management Operational research. Management science Optimization programming Random variables Robust optimization Sampling methods stochastic Stochastic models Stochastic programming Studies |
| Title | A Linear Decision-Based Approximation Approach to Stochastic Programming |
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