A Transformation-Proximal Bundle Algorithm for Multistage Adaptive Robust Optimization and Application to Constrained Robust Optimal Control

Abstract: This paper presents a novel transformation-proximal bundle algorithm for multistage adaptive robust optimization problems. By partitioning recourse decisions into state and control decisions, the proposed algorithm applies affine control policy only to state decisions and allows control decisions to be fully adaptive, thus transforming the original problem into an equivalent two-stage Adaptive Robust Optimization (ARO) problem. Importantly, this multi-to-two transformation is general enough to be employed with any two-stage ARO solution algorithms, thus opening a new avenue for a variety of multistage ARO algorithms. The proximal bundle method is developed for the resulting two-stage problem along with convergence analysis. In an inventory control application, the affine disturbance-feedback control policy suffers from a severe suboptimality with an average gap of 34.88%, while the proposed algorithm generates an average gap of merely 1.68%.