Effective Separation of Disjunctive Cuts for Convex Mixed Integer Nonlinear Programs

We describe a computationally effective method for generating disjunctive inequalities for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs, and in the limit will generate an inequality as strong as can be produced by the cut-generating nonlinear program suggested by Stubbs and Mehrotra. Using this procedure, we are able to approximately optimize over the rank one simple disjunctive closure for a wide range of convex MINLP instances. The results indicate that disjunctive inequalities have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branch-and-cut solver for convex MINLPs yields significant savings in total solution time for many instances. Overall, these results suggest that with an effective separation routine, like the one proposed here, disjunctive inequalities may be as effective for solving convex MINLPs as they have been for solving mixed-integer linear programs.

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