Integer and combinatorial optimization
Integer and combinatorial optimization
A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
Stochastic decomposition: an algorithm for two-state linear programs with recourse
Mathematics of Operations Research
A lift-and-project cutting plane algorithm for mixed 0-1 programs
Mathematical Programming: Series A and B
Mixed-integer bilinear programming problems
Mathematical Programming: Series A and B
Discrete Applied Mathematics
Mathematical Programming: Series A and B
Mathematical Programming: Series A and B
L-shaped decomposition of two-stage stochastic programs with integer recourse
Mathematical Programming: Series A and B
Dual decomposition in stochastic integer programming
Operations Research Letters
Computers and Operations Research
International Journal of Computational Science and Engineering
A general algorithm for solving two-stage stochastic mixed 0-1 first-stage problems
Computers and Operations Research
Cutting plane algorithms for solving a stochastic edge-partition problem
Discrete Optimization
Exact Solution of Large-Scale Hub Location Problems with Multiple Capacity Levels
Transportation Science
Computers and Industrial Engineering
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In this paper, we modify Benders' decomposition method by using concepts from the Reformulation-Linearization Technique (RLT) and lift-and-project cuts in order to develop an approch for solving discrete optimization problems that yield integral subproblems, such as those that arise in the case of two-stage stochastic programs with integer recourse. We first demonstrate that if a particular convex hull representation of the problem's constrained region is available when binariness is enforced on only the second-stage (or recourse) variables, then the regular Benders' algorithm is applicable. The proposed procedure is based on sequentially generating a suitable partial description of this convex hull representation as needed in the process of deriving valid Benders' cuts. The key idea is to solve the subproblems using an RLT or lift-and-project cutting plane scheme, but to generate and store the cuts as functions of the first-stage variables. Hence, we are able to re-use these cutting planes from one subproblem solution to the next simply by updating the values of the first-stage decisions. The proposed Benders' cuts also recognize these RLT or lift-and-project cuts as functions of the first-stage variables, and are hence shown to be globally valid, thereby leading to an overall finitely convergent solution procedure. Some illustrative examples are provided to elucidate the proposed approach. The focus of this paper is on developing such a finitely convergent Benders' approach for problems having 0-1 mixed-integer subproblems as in the aforementioned context of two-stage stochastic programs with integer recourse. A second part of this paper will deal with related computational experiments.