  Robust optimization is a popular paradigm for modeling and solving two- and
multi-stage decision-making problems affected by uncertainty. In many
real-world applications, the time of information discovery is
decision-dependent and the uncertain parameters only become observable after an
often costly investment. Yet, most of the literature assumes that uncertain
parameters can be observed for free and that the sequence in which they are
revealed is independent of the decision-maker's actions. To fill this gap. we
consider two- and multi-stage robust optimization problems in which part of the
decision variables control the time of information discovery. Thus, information
can be discovered (at least in part) by making strategic exploratory
investments in previous stages. We propose a novel dynamic formulation of the
problem and prove its correctness. We leverage our model to provide a solution
method inspired from the K-adaptability approximation. We reformulate the
problem as a finite mixed-integer (resp. bilinear) program if none (resp. some
of the) decision variables are real-valued. This finite program is solvable
with off-the-shelf solvers. We generalize our approach to the minimization of
piecewise linear convex functions. We demonstrate the effectiveness of our
method in terms of interpretability, optimality, and speed on synthetic
instances of the Pandora box problem, the preference elicitation problem, the
best box problem, and the R&D project portfolio optimization problem. Finally,
we evaluate it on an instance of the active preference elicitation problem used
to recommend kidney allocation policies to policy-makers at the United Network
for Organ Sharing.
