On the extreme inequalities of infinite group problems

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson [10] 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 [15], Dash and S. S. Dey School of Industrial Engineering, Purdue University. Supported by NSF Grant DMI-03-48611. E-mail: [email protected] J.-P. P. Richard School of Industrial Engineering, Purdue University. Supported by NSF Grant DMI-03-48611. E-mail: [email protected] Y. Li Krannert School of Management, Purdue University. E-mail: [email protected] L. A. Miller Department of Mechanical Engineering, University of Minnesota. E-mail: [email protected] 2 Santanu S. Dey et al. Günlük [3,4] and Richard, Li and Miller [17]. 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.