Adjustable Robust Optimization Reformulations of Two-Stage Worst-Case Regret Minimization Problems

This paper explores the idea that two-stage worst-case regret minimization problems with either objective or right-hand side uncertainty can be reformulated as two-stage robust optimization problems and can therefore benefit from the solution schemes and theoretical knowledge that have been developed in the last decade for this class of problems. In particular, we identify conditions under which a first-stage decision can be obtained either exactly using popular adjustable robust optimization decomposition schemes or approximately by conservatively using affine decision rules. Furthermore, we provide both numerical and theoretical evidence that, in practice, the first-stage decision obtained using affine decision rules is of high quality. Initially, this is done by establishing mild conditions under which these decisions can be proven exact, which effectively extends the space of regret minimization problems known to be solvable in polynomial time. We further evaluate both the suboptimality and computational efficiency of this tractable approximation scheme in a multi-item newsvendor problem and a production transportation problem.