The C 3 theorem and a D 2 algorithm for large scale stochastic integer programming

Abstract

This paper considers the two stage stochastic integer programming problems, with an emphasis on problems in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C^3 (Common Cut Coefficients) Theorem. Based on the C^3 Theorem, we develop an algorithmic methodology that we refer to as Disjunctive Decomposition (D^2). We show that when the second stage consists of 0-1 MILP problems , we can obtain accurate second stage objective function estimates afer finitely many steps. We also set the stage for comparisions between problems in which the first stage includes only 0-1 variables and those that allow both continuous and integer variables in the first stage.