  We study two-stage distributionally robust optimization (DRO) problems with
decision-dependent information discovery (DDID) wherein (a portion of) the
uncertain parameters are revealed only if an (often costly) investment is made
in the first stage. This class of problems finds many important applications in
selection problems (e.g., in hiring, project portfolio optimization, or optimal
sensor location). Despite the problem's wide applicability, it has not been
previously studied. We propose a framework for modeling and approximately
solving DRO problems with DDID. We formulate the problem as a min-max-min-max
problem and adopt the popular K-adaptability approximation scheme, which
chooses K candidate recourse actions here-and-now and implements the best of
those actions after the uncertain parameters that were chosen to be observed
are revealed. We then present a decomposition algorithm that solves the
K-adaptable formulation exactly. In particular, we devise a cutting plane
algorithm which iteratively solves a relaxed version of the problem, evaluates
the true objective value of the corresponding solution, generates valid cuts,
and imposes them in the relaxed problem. For the evaluation problem, we develop
a branch-and-cut algorithm that provably converges to an optimal solution. We
showcase the effectiveness of our framework on the R&D project portfolio
optimization problem and the best box problem.
