Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms

Abstract.We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables. We develop a lifting theory for the continuous variables. In particular, we present a pseudo-polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use.

[1]  Z. Gu,et al.  Lifted cover inequalities for 0-1 and mixed 0-1 integer programs , 1995 .

[2]  Eitan Zemel,et al.  Lifting the facets of zero–one polytopes , 1978, Math. Program..

[3]  Laurence A. Wolsey,et al.  The 0-1 Knapsack problem with a single continuous variable , 1999, Math. Program..

[4]  George L. Nemhauser,et al.  A polyhedral approach to combinatorial complementarity programming problems , 1995 .

[5]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

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

[7]  Martin W. P. Savelsbergh,et al.  Lifted flow cover inequalities for mixed 0-1 integer programs , 1999, Math. Program..

[8]  George L. Nemhauser,et al.  A family of inequalities for the generalized assignment polytope , 2001, Oper. Res. Lett..

[9]  George L. Nemhauser,et al.  A polyhedral study of the cardinality constrained knapsack problem , 2002, Math. Program..

[10]  Peter L. Hammer,et al.  Facet of regular 0–1 polytopes , 1975, Math. Program..

[11]  Egon Balas,et al.  Facets of the knapsack polytope , 1975, Math. Program..

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

[13]  E. Balas,et al.  Facets of the Knapsack Polytope From Minimal Covers , 1978 .

[14]  Laurence A. Wolsey,et al.  Faces for a linear inequality in 0–1 variables , 1975, Math. Program..

[15]  Laurence A. Wolsey,et al.  Technical Note - Facets and Strong Valid Inequalities for Integer Programs , 1976, Oper. Res..

[16]  George L. Nemhauser,et al.  Facets of the Complementarity Knapsack Polytope , 2002, Math. Oper. Res..

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

[18]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[19]  Michael Jünger,et al.  Combinatorial optimization - Eureka, you shrink! , 2003 .

[20]  Ellis L. Johnson,et al.  Solving Large-Scale Zero-One Linear Programming Problems , 1983, Oper. Res..

[21]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Computation , 1998, INFORMS J. Comput..

[22]  Michael Jünger,et al.  Combinatorial Optimization -- Eureka, You Shrink!: Papers Dedicated to Jack Edmonds. 5th International Workshop, Aussois, France, March 5-9, 2001, Revised Papers , 2003 .

[23]  George L. Nemhauser,et al.  A generalized assignment problem with special ordered sets: a polyhedral approach , 2000, Math. Program..

[24]  Martin W. P. Savelsbergh,et al.  Lifted Cover Inequalities for 0-1 Integer Programs: Complexity , 1999, INFORMS J. Comput..

[25]  George L. Nemhauser,et al.  Lifted inequalities for 0-1 mixed integer programming: Superlinear lifting , 2003, Math. Program..