Robust optimization (RO) is a popular paradigm for modeling and solving two- and multi-stage decision-making problems affected by uncertainty. In many real-world applications, such as R&D project selection, production planning, or preference elicitation for product or policy recommendations, 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 on robust optimization assumes that the 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 in the practicability of RO, we consider two- and multi-stage robust optimization problems in which part of the decision variables control the time of information discovery. Thus, information available at any given time is decision-dependent and 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, whereby K candidate strategies for each decision stage are chosen here-and-now and, at the beginning of each period, the best of these strategies is selected after the uncertain parameters that were chosen to be observed are revealed. 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 with real-valued recommendations, 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 based on real data from the U.S. Kidney Allocation System.
Technical Report, University of Southern California, September 2019