Improving the integer L-shaped method Gustavo Angulo, Shabbir Ahmed, Santanu S. Dey Georgia Institute of Technology gangulo@gatech.edu, sahmed@isye.gatech.edu, santanu.dey@isye.gatech.edu April 29, 2014 Abstract We consider the integer L-shaped method for two-stage stochastic integer programs. To improve the performance of the algorithm, we present and combine two strategies. First, to avoid time- consuming exact evaluations of the second-stage cost function, we propose a simple modification that alternates between linear and mixed-integer subproblems. Then, to better approximate the shape of the second-stage cost function, we present a general framework to generate optimality cuts via a cut-generating linear program which considers information from all solutions found up to any given stage of the method. In order to address the impact of the proposed approaches, we report computa- tional results on two classes of stochastic integer problems. 1 Introduction In this work we consider mixed-integer programs of the form (IP) min x,z,θ cx + dz + θ s.t. Ax + Cz ≤ b (1) Q( x) − θ ≤ 0 (2) x ∈{0, 1} n (3) z ≥ 0, z ∈ Z, (4) where Z is a mixed-integer set and Q( x) is a real-valued function taking a binary vector x as argument. We say that ( x ∗ , z ∗ , θ ∗ ) is a candidate solution if ( x ∗ , z ∗ ) satisfies (1), (3), and (4). If in addition (2) holds, then we say ( x ∗ , z ∗ , θ ∗ ) is a feasible (candidate) solution. Constraint (2) together with the presence of θ in the objective function ensures θ = Q( x) is satisfied by any optimal solution to (IP). A fundamental assumption is that given x, Q( x) can be computed with a reasonable amount of effort. In the context of two-stage stochastic integer programming, we usually have Q( x) := E ξ  min y {qy : Wy = h − Tx, y ∈ Y}  , which denotes the expected second-stage cost of x with respect to the random data ξ =(q, W, T, h). We assume that Y imposes some integrality requirements on y. When ξ has a finite set of possible 1