On a generalization of the master cyclic group polyhedron

AbstractWe study the master equality polyhedron (MEP) which generalizes the master cyclic group polyhedron (MCGP) and the master knapsack polyhedron (MKP). We present an explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP. This result generalizes similar results for the MCGP by Gomory (1969) and for the MKP by Araóz (1974). Furthermore, this characterization gives a polynomial time algorithm for separating an arbitrary point from MEP. We describe how facet-defining inequalities for the MCGP can be lifted to obtain facet-defining inequalities for MEP, and also present facet-defining inequalities for MEP that cannot be obtained in such a way. Finally, we study the mixed-integer extension of MEP and present an interpolation theorem that produces valid inequalities for general mixed integer programming problems using facets of MEP.

[1]  Ellis L. Johnson,et al.  Corner Polyhedra and their connection with cutting planes , 2003, Math. Program..

[2]  Laurence A. Wolsey,et al.  Technical Note - Facets and Strong Valid Inequalities for Integer Programs , 1976, Oper. Res..

[3]  E. Balas,et al.  Facets of the Knapsack Polytope From Minimal Covers , 1978 .

[4]  Ellis L. Johnson,et al.  Cyclic group and knapsack facets , 2003, Math. Program..

[5]  Matteo Fischetti,et al.  Mixed-Integer Cuts from Cyclic Groups , 2005, IPCO.

[6]  Sanjeeb Dash,et al.  On the strength of Gomory mixed-integer cuts as group cuts , 2008, Math. Program..

[7]  Ellis L. Johnson,et al.  Some continuous functions related to corner polyhedra , 1972, Math. Program..

[8]  Ellis L. Johnson,et al.  Some continuous functions related to corner polyhedra, II , 1972, Math. Program..

[9]  Alper Atamtürk,et al.  Sequence Independent Lifting for Mixed-Integer Programming , 2004, Oper. Res..

[10]  R. Gomory.  Some polyhedra related to combinatorial problems , 1969 .

[11]  Marcus Poggi de Aragão,et al.  Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation , 2007, Math. Program..

[12]  Sanjeeb Dash,et al.  Valid inequalities based on the interpolation procedure , 2006, Math. Program..

[13]  Ellis L. Johnson,et al.  T-space and cutting planes , 2003, Math. Program..

[14]  Sanjeeb Dash,et al.  Valid inequalities based on simple mixed-integer sets , 2004, Math. Program..

[15]  Matteo Fischetti,et al.  How tight is the corner relaxation? , 2008, Discret. Optim..

[16]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988, Wiley interscience series in discrete mathematics and optimization.