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1. Strong and compact relaxations in the original space using a compact extended formulation

Author

Mathieu Van Vyve and Laurence A. Wolsey

Subjects
Production Site, Extended Formulation, Decomposition Algorithm, Valid Inequality, Linear Programming Relaxation, Applied mathematics. Quantitative methods, T57-57.97, Electronic computers. Computer science, QA75.5-76.95
Abstract

For certain integer programs, one way to obtain a strong dual bound is to use an extended formulation and then solve the associated linear programming relaxation. The classical way to obtain a bound of the same value in the original variable space is through the use of Benders’ algorithm. Here, we propose an alternative approach based on a decomposition of the dual optimal solution of the extended formulation linear program. An example of the approach using the multi-commodity formulation of a two-level production/transportation problem is presented.

Published
2013
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34. Lattice Reformulation Cuts

Author

Andrea Tramontani, Frederik von Heymann, Andrea Lodi, Laurence A. Wolsey, Karen Aardal, UCL - SSH/LIDAM/CORE - Center for operations research and econometrics, and Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Subjects
Discrete mathematics, Lattice basis reduction, Integer programming, cutting planes, Theoretical Computer Science, Lattice (module), Lattice reformulations, Cutting planes, lattice basis reduction, Lattice reduction, lattice reformulations, Software, Mathematics, Integer (computer science)
Abstract

Here we consider the question whether the lattice reformulation of a linear integer program can be used to produce effective cutting planes. In particular, we aim at deriving split cuts that cut off more of the integrality gap than Gomory mixed-integer (GMI) inequalities generated from LP-tableaus, while being less computationally demanding than generating the split closure. We consider integer programs (IPs) in the form max{ Ax=b x =Zn+}, where the reformulation takes the form max\{cx +cQ> -xu u =Zn-m Z n - m\}, where Q is an n (n - m) integer matrix. Working on an optimal LP-tableau in the u -space allows us to generate n - m GMIs in addition to the m GMIs associated with the optimal tableau in the x space. These provide new cuts that can be seen as GMIs associated to n - m nonelementary split directions associated with the reformulation matrix \Q . On the other hand it turns out that the corner polyhedra associated to an LP basis and the GMI or split closures are the same whether working in the x or u spaces. Our theoretical derivations are accompanied by an illustrative computational study. The computations show that the effectiveness of the cuts generated by this approach depends on the quality of the reformulation obtained by the reduced basis algorithm used to generate Q and that it is worthwhile to generate several rounds of such cuts. However, the effectiveness of the cuts deteriorates as the number of constraints is increased.

Published
2021

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