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Title: Generating moment matching scenarios using optimization techniques

Abstract

An optimization based method is proposed to generate moment matching scenarios for numerical integration and its use in stochastic programming. The main advantage of the method is its flexibility: it can generate scenarios matching any prescribed set of moments of the underlying distribution rather than matching all moments up to a certain order, and the distribution can be defined over an arbitrary set. This allows for a reduction in the number of scenarios and allows the scenarios to be better tailored to the problem at hand. The method is based on a semi-infinite linear programming formulation of the problem that is shown to be solvable with polynomial iteration complexity. A practical column generation method is implemented. The column generation subproblems are polynomial optimization problems; however, they need not be solved to optimality. It is found that the columns in the column generation approach can be efficiently generated by random sampling. The number of scenarios generated matches a lower bound of Tchakaloff's. The rate of convergence of the approximation error is established for continuous integrands, and an improved bound is given for smooth integrands. Extensive numerical experiments are presented in which variants of the proposed method are compared to Monte Carlomore » and quasi-Monte Carlo methods on both numerical integration problems and stochastic optimization problems. The benefits of being able to match any prescribed set of moments, rather than all moments up to a certain order, is also demonstrated using optimization problems with 100-dimensional random vectors. Here, empirical results show that the proposed approach outperforms Monte Carlo and quasi-Monte Carlo based approaches on the tested problems.« less

Authors:
 [1];  [1]
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  1. Northwestern Univ., Evanston, IL (United States)
Publication Date:
Research Org.:
Northwestern Univ., Evanston, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1321131
Grant/Contract Number:  
SC0005102
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Optimization
Additional Journal Information:
Journal Volume: 23; Journal Issue: 2; Journal ID: ISSN 1052-6234
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; scenario generation; moment matching; cubature; column generation; convex programming; statistical bounds; semi-infinite programming

Citation Formats

Mehrotra, Sanjay, and Papp, Dávid. Generating moment matching scenarios using optimization techniques. United States: N. p., 2013. Web. doi:10.1137/110858082.
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Mehrotra, Sanjay, & Papp, Dávid. Generating moment matching scenarios using optimization techniques. United States. https://doi.org/10.1137/110858082
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Mehrotra, Sanjay, and Papp, Dávid. Thu . "Generating moment matching scenarios using optimization techniques". United States. https://doi.org/10.1137/110858082. https://www.osti.gov/servlets/purl/1321131.
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@article{osti_1321131,
title = {Generating moment matching scenarios using optimization techniques},
author = {Mehrotra, Sanjay and Papp, Dávid},
abstractNote = {An optimization based method is proposed to generate moment matching scenarios for numerical integration and its use in stochastic programming. The main advantage of the method is its flexibility: it can generate scenarios matching any prescribed set of moments of the underlying distribution rather than matching all moments up to a certain order, and the distribution can be defined over an arbitrary set. This allows for a reduction in the number of scenarios and allows the scenarios to be better tailored to the problem at hand. The method is based on a semi-infinite linear programming formulation of the problem that is shown to be solvable with polynomial iteration complexity. A practical column generation method is implemented. The column generation subproblems are polynomial optimization problems; however, they need not be solved to optimality. It is found that the columns in the column generation approach can be efficiently generated by random sampling. The number of scenarios generated matches a lower bound of Tchakaloff's. The rate of convergence of the approximation error is established for continuous integrands, and an improved bound is given for smooth integrands. Extensive numerical experiments are presented in which variants of the proposed method are compared to Monte Carlo and quasi-Monte Carlo methods on both numerical integration problems and stochastic optimization problems. The benefits of being able to match any prescribed set of moments, rather than all moments up to a certain order, is also demonstrated using optimization problems with 100-dimensional random vectors. Here, empirical results show that the proposed approach outperforms Monte Carlo and quasi-Monte Carlo based approaches on the tested problems.},
doi = {10.1137/110858082},
journal = {SIAM Journal on Optimization},
number = 2,
volume = 23,
place = {United States},
year = {Thu May 16 00:00:00 EDT 2013},
month = {Thu May 16 00:00:00 EDT 2013}
}
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https://doi.org/10.1137/110858082
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Works referenced in this record:

Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective
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    Scenario generation in stochastic programming using principal component analysis based on moment-matching approach
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    • European Journal of Operational Research, Vol. 249, Issue 1
    • DOI: 10.1016/j.ejor.2015.05.048
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