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57
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 48 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
A Precise Correspondence Between LiftandProject Cuts, Simple Disjunctive Cuts
 Mathematical Programming B
, 2003
"... Abstract We establish a precise correspondence between liftandproject cuts for mixed 01 programs, simple disjunctive cuts (intersection cuts) and mixedinteger Gomory cuts. The correspondence maps members of one family onto members of the others. It also maps bases of the higherdimensional cut g ..."
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Cited by 31 (4 self)
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Abstract We establish a precise correspondence between liftandproject cuts for mixed 01 programs, simple disjunctive cuts (intersection cuts) and mixedinteger Gomory cuts. The correspondence maps members of one family onto members of the others. It also maps bases of the higherdimensional cut generating linear program (CGLP) into bases of the linear programming relaxation. It provides new bounds on the number of facets of the elementary closure, and on the rank, of the standard linear programming relaxation of the mixed 01 polyhedron with respect to the above families of cutting planes.
Valid inequalities for mixed integer linear programs
 Mathematical Programming B
, 2006
"... Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mi ..."
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Cited by 31 (0 self)
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Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Key words: mixed integer linear program, liftandproject, split cut, Gomory cut, mixed integer rounding, elementary closure, polyhedra, union of polyhedra 1.
Minimal Valid Inequalities for Integer Constraints
, 2007
"... Dedicated to George Nemhauser for his 70th birthday In this note we consider an infinite relaxation of mixed integer linear programs. We show that any minimal valid inequality for this infinite relaxation arises from a nonnegative, piecewise linear, convex and homogeneous function. 1 ..."
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Cited by 25 (17 self)
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Dedicated to George Nemhauser for his 70th birthday In this note we consider an infinite relaxation of mixed integer linear programs. We show that any minimal valid inequality for this infinite relaxation arises from a nonnegative, piecewise linear, convex and homogeneous function. 1
Sequence Independent Lifting for MixedInteger Programming
 Operations Research
, 2002
"... We show that superadditive lifting functions lead to sequence independent lifting of inequalities for general mixedinteger programming. ..."
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Cited by 22 (11 self)
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We show that superadditive lifting functions lead to sequence independent lifting of inequalities for general mixedinteger programming.
Elementary Closures for Integer Programs
 Operations Research Letters
, 2000
"... In integer programming, the elementary closure associated with a family of cuts is the convex set defined by the intersection of all the cuts in the family. In this paper, we compare the elementary closures arising from several classical families of cuts: three versions of Gomory's fractional cuts, ..."
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Cited by 21 (3 self)
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In integer programming, the elementary closure associated with a family of cuts is the convex set defined by the intersection of all the cuts in the family. In this paper, we compare the elementary closures arising from several classical families of cuts: three versions of Gomory's fractional cuts, three versions of Gomory's mixed integer cuts, two versions of intersection cuts and their strengthened forms, Chvátal cuts, MIR cuts, liftandproject cuts without and with strengthening, twoversions of disjunctive cuts, SheraliAdams cuts and LovászSchrijver cuts with positive semidefiniteness constraints.
On the Separation of Split Cuts and Related Inequalities
 Mathematical Programming
"... The split cuts of Cook, Kannan and Schrijver are generalpurpose valid inequalities for integer programming which include a variety of other wellknown cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was ..."
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Cited by 20 (1 self)
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The split cuts of Cook, Kannan and Schrijver are generalpurpose valid inequalities for integer programming which include a variety of other wellknown cuts as special cases. To detect violated split cuts, one has to solve the associated separation problem. The complexity of split cut separation was recently cited as an open problem by Cornuejols & Li [10]. In this paper we settle this question by proving strong NPcompleteness of separation for split cuts. As a byproduct we also show NPcompleteness of separation for several other classes of inequalities, including the MIRinequalities of Nemhauser and Wolsey and some new inequalities which we call balanced split cuts and binary split cuts. We also strengthen NPcompleteness results of Caprara & Fischetti [5] (for {0, 1 2 }cuts) and Eisenbrand [12] (for ChvatalGomory cuts). To compensate for this bleak picture, we also give a positive result for the Symmetric Travelling Salesman Problem. We show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle. Key words: Cutting planes, separation, complexity, travelling salesman problem, comb inequalities. 1
Lifting Integer Variables in Minimal Inequalities Corresponding To LatticeFree Triangles
"... Recently, Andersen et al. [1] and Borozan and Cornuéjols [3] characterized the minimal inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal latticefree convex se ..."
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Cited by 19 (2 self)
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Recently, Andersen et al. [1] and Borozan and Cornuéjols [3] characterized the minimal inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal latticefree convex sets. In order to use these minimal inequalities to obtain cuts from two rows of a general simplex tableau, it is necessary to extend the system to include integer variables (giving the twodimensional mixed integer infinite group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we analyze the lifting of minimal inequalities derived from latticefree triangles. Maximal latticefree triangles in R 2 can be classified into three categories: those with multiple integral points in the relative interior of one of its sides, those with integral vertices and one integral point in the relative interior of each side, and those with non integral vertices and one integral point in the relative interior of each side. We prove that the lifting functions are unique for each of the first two categories such that the resultant inequality is minimal for the mixed integer infinite group problem, and characterize them. We show that the lifting function is not
Mixing MixedInteger Inequalities
 MATHEMATICAL PROGRAMMING
, 1998
"... Mixedinteger rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixedinteger programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities. Given a mixedinteger region S and a coll ..."
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Cited by 18 (2 self)
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Mixedinteger rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixedinteger programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities. Given a mixedinteger region S and a collection of valid "base" mixedinteger inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or "mixing" them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a mixedinteger region associated with the base inequalities. We also study some extensions of this mixing procedure, and discuss how it can be used to obtain new classes of strong valid inequalities for various mixedinteger programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems.
Inequalities from Two Rows of a Simplex Tableau
, 2007
"... In this paper we explore the geometry of the integer points in a cone rooted at a rational point. This basic geometric object allows us to establish some links between lattice point free bodies and the derivation of inequalities for mixed integer linear programs by considering two rows of a simplex ..."
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Cited by 12 (2 self)
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In this paper we explore the geometry of the integer points in a cone rooted at a rational point. This basic geometric object allows us to establish some links between lattice point free bodies and the derivation of inequalities for mixed integer linear programs by considering two rows of a simplex tableau simultaneously.