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Title: Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets

Abstract

We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Furthermore, our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derivedmore » under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.« less

Authors:
 [1]; ORCiD logo [1]
  1. Univ. of Michigan, Ann Arbor, MI (United States)
Publication Date:
Research Org.:
Univ. of Michigan, Ann Arbor, MI (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
OSTI Identifier:
1735792
Alternate Identifier(s):
OSTI ID: 1995053
Grant/Contract Number:  
SC0018018; 1727618; 1709094
Resource Type:
Accepted Manuscript
Journal Name:
Mathematical Programming
Additional Journal Information:
Journal Volume: 196; Journal Issue: 1; Journal ID: ISSN 0025-5610
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Multistage sequential decision-making; Distributionally robust optimization; Endogenous uncertainty; Mixed-integer semidefinite/linear programming; Stochastic dual dynamic integer programming (SDDiP)

Citation Formats

Yu, Xian, and Shen, Siqian. Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets. United States: N. p., 2020. Web. doi:10.1007/s10107-020-01580-4.
Yu, Xian, & Shen, Siqian. Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets. United States. https://doi.org/10.1007/s10107-020-01580-4
Yu, Xian, and Shen, Siqian. Wed . "Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets". United States. https://doi.org/10.1007/s10107-020-01580-4. https://www.osti.gov/servlets/purl/1735792.
@article{osti_1735792,
title = {Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets},
author = {Yu, Xian and Shen, Siqian},
abstractNote = {We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Furthermore, our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derived under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.},
doi = {10.1007/s10107-020-01580-4},
journal = {Mathematical Programming},
number = 1,
volume = 196,
place = {United States},
year = {Wed Oct 21 00:00:00 EDT 2020},
month = {Wed Oct 21 00:00:00 EDT 2020}
}

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