An Introduction to the Theory of Cutting-Planes

Publisher Summary This chapter presents an introduction to cutting-plane theory. Two principles for obtaining cutting-planes are discussed, one for S=S1= {x ≥ 0 | Ax ≥ b} and one for S=S2= {x ≥ 0 | x integer and Ax=b}. The principle for S1 is a restatement of the duality theorem of Linear Programming. The chapter presents two metaprinciples for combining cuts obtained by the two principles, under various special circumstances. By using the principle for S=S1 together with one of the metaprinciples, the basic principle of disjunctive constraints, which has an interesting converse, is obtained. Several examples of how to use the principles and metaprinciples for producing cutting-planes for a wide variety of situations are presented. The chapter also presents several additional results of cutting-plane theory, which give a general idea of what kinds of theorems are proved by developing the basic principles further.

[1]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

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

[3]  R E Gomory,et al.  ON THE RELATION BETWEEN INTEGER AND NONINTEGER SOLUTIONS TO LINEAR PROGRAMS. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Fred W. Glover,et al.  Concave Programming Applied to a Special Class of 0-1 Integer Programs , 1973, Oper. Res..

[5]  G. Dantzig.  Note on solving linear programs in integers , 1959 .

[6]  M. Fisher,et al.  Constructive Duality in Integer Programming , 1974 .

[7]  Fred W. Glover,et al.  The Generalized Lattice-Point Problem , 1973, Oper. Res..

[8]  Robert J. Townsley,et al.  THE MAXIMIZATION OF A QUADRATIC FUNCTION OF VARIABLES SUBJECT TO LINEAR INEQUALITIES*t , 1964 .

[9]  B. Eaves.  The Linear Complementarity Problem , 1971 .

[10]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[11]  Charles E. Blair,et al.  A converse for disjunctive constraints , 1978 .

[12]  Charles Eugene Blair.  Two Rules for Deducing Valid Inequalities for 0-1 Problems , 1976 .

[13]  Egon Balas,et al.  Integer programming and convex analysis: Intersection cuts from outer polars , 1972, Math. Program..

[14]  Jeremy F. Shapiro,et al.  Generalized Lagrange Multipliers in Integer Programming , 2011, Oper. Res..

[15]  Robert R. Meyer,et al.  On the existence of optimal solutions to integer and mixed-integer programming problems , 1974, Math. Program..

[16]  Fred W. Glover,et al.  Convexity Cuts and Cut Search , 1973, Oper. Res..

[17]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

[18]  Claude-Alain Burdet.  Enumerative inequalities in integer programming , 1972, Math. Program..

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

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

[21]  Robert G. Jeroslow,et al.  Cutting-Plane Theory: Disjunctive Methods , 1977 .

[22]  Robert G. Jeroslow,et al.  Trivial integer programs unsolvable by branch-and-bound , 1974, Math. Program..

[23]  A. M. Geoffrion,et al.  Integer Programming Algorithms: A Framework and State-of-the-Art Survey , 1972 .

[24]  G. Owen.  Cutting planes for programs with disjunctive constraints , 1973 .

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

[26]  R E Gomory,et al.  Faces of an integer polyhedron. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[27]  G. Dantzig,et al.  COMPLEMENTARY PIVOT THEORY OF MATHEMATICAL PROGRAMMING , 1968 .

[28]  Egon Balas,et al.  Intersection Cuts - A New Type of Cutting Planes for Integer Programming , 1971, Oper. Res..

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

[30]  Robert G. Jeroslow,et al.  Some Basis Theorems for Integral Monoids , 1978, Math. Oper. Res..

[31]  Fred W. Glover,et al.  Polyhedral annexation in mixed integer and combinatorial programming , 1975, Math. Program..

[32]  Laurence A. Wolsey.  Group-Theoretic Results in Mixed Integer Programming , 1971, Oper. Res..

[33]  G. Dantzig.  Discrete-Variable Extremum Problems , 1957 .

[34]  Peter L. Hammer,et al.  Pseudo-Boolean Programming , 1969, Oper. Res..

[35]  R. D. Young.  Hypercylindrically Deduced Cuts in Zero-One Integer Programs , 1971, Oper. Res..

[36]  D. Hilbert.  Über die Theorie der algebraischen Formen , 1890 .