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36
Valid inequalities for mixed integer linear programs
 MATHEMATICAL PROGRAMMING B
, 2006
"... 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 integ ..."
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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.
Optimizing over the split closure
 CARNEGIE MELLON UNIVERSITY
, 2006
"... The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LPrelaxation P of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of P. This paper deals with the problem of optimizing over the split closure. This is ..."
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Cited by 38 (3 self)
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The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LPrelaxation P of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of P. This paper deals with the problem of optimizing over the split closure. This is accomplished by repeatedly solving the following separation problem: given a fractional point, say x, find a rank1 split cut violated by x or show that none exists. We show that this separation problem can be formulated as a parametric mixed integer linear program (PMILP) with a single parameter in the objective function and the right hand side. We develop an algorithmic framework to deal with the resulting PMILP by creating and maintaining a dynamically updated grid of parameter values, and use the corresponding mixed integer programs to generate rank 1 split cuts. Our approach was implemented in the COINOR framework using CPLEX 9.0 as a general purpose MIP solver. We report our computational results on wellknown benchmark instances from MIPLIB 3.0 and Capacitated Warehouse Location Problems from OrLib. Our computational results show that rank1 split cuts close more than 98 % of the duality
Elementary closures for integer programs
 OPERATIONS RESEARCH LETTERS
, 2001
"... In integer programming, the elementary closure associated with a family of cuts is the convex set dened 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 cut ..."
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Cited by 24 (3 self)
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In integer programming, the elementary closure associated with a family of cuts is the convex set dened 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, Chvatal cuts, MIR cuts, liftandproject cuts without and with strengthening, two versions of disjunctive cuts, SheraliAdams cuts and LovaszSchrijver cuts with positive semideniteness constraints.
On the rank of mixed 0,1 polyhedra
, 2001
"... For a polytope in the [0; 1] n cube, Eisenbrand and Schulz showed recently that the maximum Chvatal rank is bounded above by O(n² logn) and bounded below by (1 + )n for some > 0. Chvatal cuts are equivalent to Gomory fractional cuts, which are themselves dominated by Gomory mixed integer cuts. Wh ..."
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Cited by 14 (3 self)
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For a polytope in the [0; 1] n cube, Eisenbrand and Schulz showed recently that the maximum Chvatal rank is bounded above by O(n² logn) and bounded below by (1 + )n for some > 0. Chvatal cuts are equivalent to Gomory fractional cuts, which are themselves dominated by Gomory mixed integer cuts. What do these upper and lower bounds become when the rank is dened relative to Gomory mixed integer cuts? An upper bound of n follows from existing results in the literature. In this note, we show that the lower bound is also equal to n. This result still holds for mixed 0,1 polyhedra with n binary variables.
On the Separation of Disjunctive Cuts
 MATHEMATICAL PROGRAMMING
"... Disjunctive cuts for MixedInteger Linear Programs have been introduced by Egon Balas in the late 70’s, and successfully exploited in practice since the late 90’s. In this paper we investigate the main ingredients of a disjunctive cut separation procedure, and analyze their impact on the quality o ..."
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Cited by 13 (4 self)
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Disjunctive cuts for MixedInteger Linear Programs have been introduced by Egon Balas in the late 70’s, and successfully exploited in practice since the late 90’s. In this paper we investigate the main ingredients of a disjunctive cut separation procedure, and analyze their impact on the quality of the rootnode bound for a set of instances taken from MIPLIB library. We compare alternative normalization conditions, and try to better understand their role. In particular we point out that the constraints that become redundant (because of the disjunction used) can produce overweak cuts, and analyze this property with respect to the normalization used. Finally, we introduce a new normalization condition and analyze its theoretical properties and computational behavior. Along the paper, we make use of a number of small numerical examples to illustrate some basic (and often misinterpreted) disjunctive programming features.
Projection, lifting and extended formulation in integer and combinatorial optimization
 ANNALS OF OPERATIONS RESEARCH
, 2005
"... This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in di ..."
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This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like projection and restriction, the integralitypreserving property of projection, the dimension of projected polyhedra, conditions for facets of a polyhedron to project into facets of its projections, and so on. The next two sections describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations or lifting. Section 5 deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 01 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 01 programs, with the related concept of disjunctive rank. Section 6 reviews liftandproject cuts, the construction of cut generating linear programs, and techniques for lifting and for strengthening disjunctive cuts. Section 7 discusses the recently discovered possibility of solving the higher dimensional cut generating linear program without explicitly constructing it, by a sequence of properly chosen pivots in the simplex tableau of the linear programming relaxation. Finally, section 8 deals with different ways of combining cuts with branch and bound, and briefly discusses computational experience with liftandproject cuts.
Split closure and intersection cuts
 MATH. PROGRAM., SER. A 102: 457–493 (2005)
, 2005
"... In the seventies, Balas introduced intersection cuts for a Mixed Integer Linear Program (MILP), and showed that these cuts can be obtained by a closed form formula from a basis of the standard linear programming relaxation. In the early nineties, Cook, Kannan and Schrijver introduced the split closu ..."
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Cited by 11 (5 self)
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In the seventies, Balas introduced intersection cuts for a Mixed Integer Linear Program (MILP), and showed that these cuts can be obtained by a closed form formula from a basis of the standard linear programming relaxation. In the early nineties, Cook, Kannan and Schrijver introduced the split closure of a MILP, and showed that the split closure is a polyhedron. In this paper, we show that the split closure can be obtained using only intersection cuts. We give two different proofs of this result, one geometric and one algebraic. The result is then used to provide a new proof of the fact that the split closure of a MILP is a polyhedron. Finally, we extend the result to more general disjunctions.
Effective Separation of Disjunctive Cuts for Convex Mixed Integer Nonlinear Programs
, 2010
"... We describe a computationally effective method for generating disjunctive inequalities for convex mixedinteger nonlinear programs (MINLPs). The method relies on solving a sequence of cutgenerating linear programs, and in the limit will generate an inequality as strong as can be produced by the cut ..."
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We describe a computationally effective method for generating disjunctive inequalities for convex mixedinteger nonlinear programs (MINLPs). The method relies on solving a sequence of cutgenerating linear programs, and in the limit will generate an inequality as strong as can be produced by the cutgenerating nonlinear program suggested by Stubbs and Mehrotra. Using this procedure, we are able to approximately optimize over the rank one simple disjunctive closure for a wide range of convex MINLP instances. The results indicate that disjunctive inequalities have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branchandcut solver for convex MINLPs yields significant savings in total solution time for many instances. Overall, these results suggest that with an effective separation routine, like the one proposed here, disjunctive inequalities may be as effective for solving convex MINLPs as they have been for solving mixedinteger linear programs.
Liftandproject cuts for mixed integer convex programs
 Proceedings of the 15th IPCO Conference, volume 6655 of Lecture
"... Abstract. This paper addresses the problem of generating cuts for mixed integer nonlinear programs where the objective is linear and the relations between the decision variables are described by convex functions defining a convex feasible region. We propose a new method for strengthening the contin ..."
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Abstract. This paper addresses the problem of generating cuts for mixed integer nonlinear programs where the objective is linear and the relations between the decision variables are described by convex functions defining a convex feasible region. We propose a new method for strengthening the continuous relaxations of such problems using cutting planes. Our method can be seen as a practical implementation of the liftandproject technique in the nonlinear case. To derive each cut we use a combination of a nonlinear programming subproblem and a linear outer approximation. One of the main features of the approach is that the subproblems solved to generate cuts are typically not more complicated than the original continuous relaxation. In particular they do not require the introduction of additional variables or nonlinearities. We propose several strategies for using the technique and present preliminary computational evidence of its practical interest. In particular, the cuts allow us to improve over the state of the art branchandbound of the solver Bonmin, solving more problems in faster computing times on average.