On the extreme inequalities of infinite group problems

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Günlük (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

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

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

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

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

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

[6]  Santanu S. Dey.  Two families of facets for the two-dimensional mixed integer in?nite group problem , 2006 .

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

[8]  Yanjun Li,et al.  Valid inequalities for MIPs and group polyhedra from approximate liftings , 2009, Math. Program..

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

[10]  Sanjeeb Dash,et al.  Valid inequalities based on simple mixed-integer sets , 2006, Math. Program..

[11]  Santanu S. Dey.  Strong cutting planes for unstructured mixed integer programs using multiple constraints , 2007 .

[12]  Jean-Philippe P. Richard,et al.  Sequential-Merge Facets for Two-Dimensional Group Problems , 2007, IPCO.

[13]  Jean-Philippe P. Richard,et al.  Facets of Two-Dimensional Infinite Group Problems , 2008, Math. Oper. Res..

[14]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[15]  Jean-Philippe P. Richard,et al.  New inequalities for finite and infinite group problems from approximate lifting , 2008 .

[16]  Ellis L. Johnson.  On the group problem for mixed integer programming , 1974 .

[17]  Andrea Lodi,et al.  Strengthening Chvátal-Gomory cuts and Gomory fractional cuts , 2002, Oper. Res. Lett..