On the MIR Closure of Polyhedra

We study the mixed-integer rounding (MIR) closure of polyhedra. The MIR closure of a polyhedron is equal to its split closure and the associated separation problem is NP-hard. We describe a mixed-integer programming (MIP) model with linear constraints and a non-linear objective for separating an arbitrary point from the MIR closure of a given mixed-integer set. We linearize the objective using additional variables to produce a linear MIP model that solves the separation problem approximately, with an accuracy that depends on the number of additional variables used. Our analysis yields a short proof of the result of Cook, Kannan and Schrijver (1990) that the split closure of a polyhedron is again a polyhedron. We also present some computational results with our approximate separation model.

[1]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[2]  Gérard Cornuéjols,et al.  A note on the MIR closure , 2008, Oper. Res. Lett..

[3]  Friedrich Eisenbrand,et al.  NOTE – On the Membership Problem for the Elementary Closure of a Polyhedron , 1999, Comb..

[4]  Laurence A. Wolsey,et al.  Aggregation and Mixed Integer Rounding to Solve MIPs , 2001, Oper. Res..

[5]  Juan Pablo Vielma.  A constructive characterization of the split closure of a mixed integer linear program , 2007, Oper. Res. Lett..

[6]  Egon Balas,et al.  Gomory cuts revisited , 1996, Oper. Res. Lett..

[7]  Matteo Fischetti,et al.  Projected Chvátal–Gomory cuts for mixed integer linear programs , 2008, Math. Program..

[8]  Gérard Cornuéjols,et al.  On the Rank of Mixed 0, 1 Polyhedra , 2001, IPCO.

[9]  R. Gomory.  AN ALGORITHM FOR THE MIXED INTEGER PROBLEM , 1960 .

[10]  Hanif D. Sherali,et al.  Disjunctive Programming , 2009, Encyclopedia of Optimization.

[11]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[12]  William J. Cook,et al.  Chvátal closures for mixed integer programming problems , 1990, Math. Program..

[13]  Egon Balas,et al.  Optimizing over the split closure , 2008, Math. Program..

[14]  Laurence A. Wolsey,et al.  A recursive procedure to generate all cuts for 0–1 mixed integer programs , 1990, Math. Program..

[15]  Alberto Caprara,et al.  On the separation of split cuts and related inequalities , 2003, Math. Program..

[16]  Gérard Cornuéjols,et al.  Elementary closures for integer programs , 2001, Oper. Res. Lett..

[17]  G. Nemhauser,et al.  Integer Programming , 2020 .

[18]  Matteo Fischetti,et al.  Optimizing over the first Chvátal closure , 2005, Math. Program..

[19]  Kent Andersen,et al.  Split closure and intersection cuts , 2002, Math. Program..