Browsing by Author "Savard, Gilles"

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    A Branch and Cut Algorithm for Nonconvex Quadratically Constrained Quadratic Programming
    (1999-01) Audet, Charles; Hansen, Pierre; Jaumard, Brigitte; Savard, Gilles
    We present a branch and cut algorithm that yields in finite time, a globally epsilon-optimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a branching tree using Reformulation-Linearization Techniques (RLT). To do so, four classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree, and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at any node of the tree is flexible enough to be used at other nodes. Computational results are reported. Some problems of the literature are solved, for the first time with a proof of global optimality.
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    Concavity Cuts for Disjoint Bilinear Programming
    (1999-09) Alarie, Stéphane; Audet, Charles; Jaumard, Brigitte; Savard, Gilles
    We pursue the study of concavity cuts for the disjoint bilinear programming problem. This optimization problem has two equivalent symmetric linear maxmin reformulations, leading to two sets of concavity cuts. We first examine the depth of these cuts by considering the assumptions on the boundedness of the feasible regions of both maxmin and bilinear formulations. We next propose a branch and bound algorithm which makes use of concavity cuts. We also present a procedure that eliminates degenerate solutions. Extensive computational experiences are reported. Sparse problems with up to 500 variables in each disjoint set and 100 constraints, and dense problems with up to 60 variables again in each set and 60 constraints are solved in reasonable computing times.