Abstract
In this paper, we consider a generic class of adaptive optimization problems under uncertainty, and develop a data-driven paradigm of adaptive probabilistic robust optimization (APRO) in a robust and computationally tractable manner. The paradigm comprises two phases: 1) bilayer information granulation (IG), which involves the data-mining techniques and nested decomposition of convex sets that establish and restructure the knowledge from data and 2) robustization and optimization over the restructured knowledge by the IG, which forms the APRO model. The tradeoff between the solution optimality and the robustness of the resulting data-driven APRO model can be achieved by adjusting the number of clusters and the number of nested decomposition units of the IG process. We draw the connections of the APRO model with the stochastic programming and the regular robust optimization models, respectively, and show that the APRO model can be regarded as a generalized version of both models. We show that the APRO model can be transformed into the second-order conic programming which is computationally tractable and can be solved efficiently by the off-the-shelf solvers. Furthermore, the model can be extended by robustizing the probability parameters. Finally, an application on two-stage facility location planning is presented, and the computational results demonstrate the performance and the insights of using the data-driven APRO models.