  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
(Delage and Ye, 2010) 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. 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.
