TY - JOUR
T1 - On the extreme inequalities of infinite group problems
AU - Dey, Santanu S.
AU - Richard, Jean Philippe P.
AU - Li, Yanjun
AU - Miller, Lisa A.
PY - 2010/1
Y1 - 2010/1
N2 - 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.
AB - 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.
KW - Cutting planes
KW - Extreme functions
KW - Extreme inequalities
KW - Infinite group problem
KW - Mixed integer programming
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U2 - 10.1007/s10107-008-0229-6
DO - 10.1007/s10107-008-0229-6
M3 - Article
AN - SCOPUS:67349148915
SN - 0025-5610
VL - 121
SP - 145
EP - 170
JO - Mathematical Programming
JF - Mathematical Programming
IS - 1
ER -