Oracle-Based Algorithms for Binary Two-Stage Robust Optimization

Abstract

In this work we study binary two-stage robust optimization problems with objective uncertainty. The concept of two-stage robustness is tailored for problems under uncertainty which have two different kinds of decision variables, first-stage decisions which have to be made here-and-now and second-stage decisions which can be determined each time after an uncertain scenario occured.We adapt an oracle-based algorithm, originally introduced to solve binary min-max-min robust optimization problems, to efficiently calculate lower bounds for the binary two-stage robust problem by using an oracle for the underlying deterministic problem. We show that the latter lower bound can be implemented in a branch & bound procedure, where the branching is performed only over the first-stage decision variables. Additionally we show that the same procedure can be extended for non-linear binary problems which can be linearized. As an alternative solution method we apply a famous column-andconstraint generation algorithm to the binary two-stage robust problem with objective uncertainty.We test both algorithms on benchmark instances of the uncapacitated single-allocation hub-location problem, which is classically modeled by a quadratic binary formulation and show that the branch & bound procedure outperforms the column-and-constraint generation algorithm.