A Hierarchy of Bounds for Stochastic Mixed-Integer Programs

Strong relaxations are critical for solving deterministic mixed-integer programs. As solving stochastic mixed-integer programs (SMIPs) is even harder, it is likely that strong relaxations will also prove essential for SMIPs. We consider general two-stage SMIPs with recourse, where integer variables are allowed in both stages of the problem and randomness is allowed in the objective function, the constraint matrices (i.e., the technology matrix and the recourse matrix), and the right-hand side. We develop a hierarchy of lower and upper bounds for the optimal objective value of an SMIP by generalizing the wait-and-see (WS) solution and the expected result of using the expected value (EEV) solution. These bounds become progressively stronger but, generally, more difficult to compute. Our numerical study indicates that the bounds developed in this paper can be very strong relative to those provided by stochastic linear programming relaxations