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Small and Strong Formulations for Unions of Convex Sets from the Cayley Embedding

Juan Pablo Vielma(jvielma***at***mit.edu)

Abstract: There is often a significant trade-off between formulation strength and size in mixed integer programming (MIP). When modelling convex disjunctive constraints (e.g. unions of convex sets) this trade-off can be resolved by adding auxiliary continuous variables. However, adding these variables can result in a deterioration of the computational effectiveness of the formulation. For this reason, there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables. We introduce a technique to construct formulations without these detrimental continuous auxiliary variables. To develop this technique we introduce a natural non-polyhedral generalization of the Cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding. We then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints. In particular, we show it can recover and generalize all known strong formulations without continuous auxiliary variables.

Keywords: Mixed integer nonlinear programming; Mixed integer programming formulations; Disjunctive constraints

Category 1: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 2: Integer Programming (0-1 Programming )


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Entry Submitted: 04/12/2017
Entry Accepted: 04/13/2017
Entry Last Modified: 04/12/2017

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