  Two-stage stochastic mixed-integer programming (SMIP) problems with general
integer variables in the second-stage are generally difficult to solve. This
paper develops the theory of integer set reduction for characterizing the
subset of the convex hull of feasible integer points of the second-stage
subproblem which can be used for solving the SMIP. The basic idea is to
consider a small enough subset of feasible integer points that is necessary for
generating a valid inequality for the integer subproblem. An algorithm for
obtaining such a subset based on the solution of the subproblem LP-relaxation
is then devised and incorporated into the Fenchel decomposition method for
SMIP. To demonstrate the performance of the new integer set reduction
methodology, a computational study based on randomly generated test instances
was performed. The results of the study show that integer set reduction
provides significant gains in terms of generating cuts faster leading to better
bounds in solving SMIPs than using a direct solver.
