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

Yu, Xian; Shen, Siqian

doi:10.1007/s10107-020-01580-4

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

Journal Article · Wed Oct 21 00:00:00 EDT 2020 · Mathematical Programming

DOI:https://doi.org/10.1007/s10107-020-01580-4· OSTI ID:1735792

Yu, Xian ^[1]; ^[2]

Univ. of Michigan, Ann Arbor, MI (United States); University of Michigan, Ann Arbor, MI
Univ. of Michigan, Ann Arbor, MI (United States)

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.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Michigan, Ann Arbor, MI (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation

Grant/Contract Number:: SC0018018

OSTI ID:: 1735792

Alternate ID(s):: OSTI ID: 1995053

Journal Information:: Mathematical Programming, Journal Name: Mathematical Programming Journal Issue: 1 Vol. 196; ISSN 0025-5610

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

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