On Distributionally Robust Multistage Convex Optimization: New Algorithms and Complexity Analysis
Shixuan Zhang, Xu Andy Sun
TL;DR
This paper proposes a new class of algorithms for solving DR-MCO, namely a sequential dual dynamic programming (Seq-DDP) algorithm and its nonsequential version (NDDP), and provides a thorough complexity analysis of the new algorithms, proving both upper complexity bounds and a matching lower bound.
Abstract
This paper presents an algorithmic study and complexity analysis for solving distributionally robust multistage convex optimization (DR-MCO). We generalize the usual consecutive dual dynamic programming (DDP) algorithm to DR-MCO and propose a new nonconsecutive DDP algorithm that explores the stages in an adaptive fashion. We introduce dual bounds in the DDP recursions to prevent the growth of Lipschitz constants of the dual approximations caused by recursive cutting plane methods. We then provide a thorough subproblem-oracle-based complexity analysis of the proposed algorithms, proving both upper complexity bounds and a matching lower bound. To the best of our knowledge, this is the first nonasymptotic complexity result for DDP-type algorithms on DR-MCO, which reveals that in some practical settings the new nonconsecutive DDP algorithm scales linearly with respect to the number of stages. Numerical examples are given to show the effectiveness of the proposed nonconsecutive DDP algorithm and the dual-bounding technique, including the reduction of the computation time or the number of subproblem oracle evaluations, and the capability to solve problems without relatively complete recourse.