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On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of s...

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Main Authors: Thuy Anh Ta, Tien Mai, Fabian Bastin, Pierre L’Ecuyer
Format: Bài trích
Language:eng
Published: Mathematical Programming 2021
Subjects:
Sample average approximation
Multistage stochastic program
Expected value constraints
Online Access:https://link.springer.com/article/10.1007/s10107-020-01518-w
https://dlib.phenikaa-uni.edu.vn/handle/PNK/2857
https://doi.org/10.1007/s10107-020-01518-w
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spelling oai:localhost:PNK-28572022-08-17T05:54:48Z On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling Thuy Anh Ta Tien Mai Fabian Bastin Pierre L’Ecuyer Sample average approximation Multistage stochastic program Expected value constraints We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it. 2021-09-14T07:14:54Z 2021-09-14T07:14:54Z 2021 Bài trích https://link.springer.com/article/10.1007/s10107-020-01518-w https://dlib.phenikaa-uni.edu.vn/handle/PNK/2857 https://doi.org/10.1007/s10107-020-01518-w eng Mathematical Programming
institution Digital Phenikaa
collection Digital Phenikaa
language eng
topic Sample average approximation
Multistage stochastic program
Expected value constraints
spellingShingle Sample average approximation
Multistage stochastic program
Expected value constraints
Thuy Anh Ta
Tien Mai
Fabian Bastin
Pierre L’Ecuyer
On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling
description We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it.
format Bài trích
author Thuy Anh Ta
Tien Mai
Fabian Bastin
Pierre L’Ecuyer
author_facet Thuy Anh Ta
Tien Mai
Fabian Bastin
Pierre L’Ecuyer
author_sort Thuy Anh Ta
title On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling
title_short On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling
title_full On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling
title_fullStr On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling
title_full_unstemmed On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling
title_sort on a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling
publisher Mathematical Programming
publishDate 2021
url https://link.springer.com/article/10.1007/s10107-020-01518-w
https://dlib.phenikaa-uni.edu.vn/handle/PNK/2857
https://doi.org/10.1007/s10107-020-01518-w
_version_ 1751856266295115776
score 8.891145

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