A note on discrete robust optimization without the full recourse property

We denote, for all feasible x, by R(x) the associated robust problem when x is fixed, and we denote by R(x, d) the associated robust problem when x and d are fixed. In this note we assume that the deterministic problem doesn’t satisfied the full recourse property. We prove how to obtain a new MILP denoted (P̃ ) such that the robust associated problem has the same optimal solution as the initial robust problem and (P̃ ) satisfies the full recourse property. We only assume that the robust problem (PR) has a finite optimal solution, i.e. there exists M such that v(PR) ≤ M. In addition, the method detects if the problem has no solutions. To obtain (P̃ ), we add new recourse variables wt, t = 1, ..., T. Let ε be a given strictly positive value, we define the following MILP: