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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.
An Integer Programming Approach for Linear Programs with Probabilistic Constraints
, 2008
"... Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the righthand side is random and this random vector has a finite distribution. We give a mixedinteger programming formulati ..."
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Cited by 43 (8 self)
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Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the righthand side is random and this random vector has a finite distribution. We give a mixedinteger programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results which indicate that by using our strengthened formulations, instances that are considerably larger than have been considered before can be solved to optimality.
A BranchandCut Algorithm for Capacitated Network Design Problems
 MATHEMATICAL PROGRAMMING
, 1998
"... We present a branchandcut algorithm to solve capacitated network design problems. Given a capacitated network and pointtopoint traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study ..."
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Cited by 40 (2 self)
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We present a branchandcut algorithm to solve capacitated network design problems. Given a capacitated network and pointtopoint traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixedinteger programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixedinteger rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called "knapsack branching". We also report on our computational experience using reallife data.
Extended Formulations in Combinatorial Optimization
, 2009
"... This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By ”perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequ ..."
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Cited by 29 (1 self)
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This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By ”perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called ”compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, MinkowskiWeyl’s theorem, Balas ’ theorem for the union of polyhedra, Yannakakis ’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans ’ result on the size of an extended formulation for the permutahedron, and the FaenzaKaibel extended formulation for orbitopes. Supported by the Progetto di Eccellenza 20082009 of the Fondazione Cassa di Risparmio di Padova e
A note on the split rank of intersection cuts
 Mathematical Programming
, 2009
"... In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the latticefree convex set that is used to generate the intersection cut is constructed. We call this subset the restricte ..."
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Cited by 21 (3 self)
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In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the latticefree convex set that is used to generate the intersection cut is constructed. We call this subset the restricted latticefree set. It is then shown that ⌈log 2(l) ⌉ is a lower bound on the split rank of the intersection cut, where l is the number of integer points lying on the boundary of the restricted latticefree set satisfying the condition that no two points lie on the same facet of the restricted latticefree set. The use of this result is illustrated to obtain a lower bound of ⌈log 2(n + 1) ⌉ on the split rank of nrow mixing inequalities. Over the years, many classes of cuts have been proposed for solving unstructured mixed integer programs that can be used within the branchandcut framework; see Nemhauser and Wolsey [29], Marchand, Martin, Weismantel and Wolsey [27] and Johnson, Nemhauser and Savelsbergh [25]. Among the many classes of cutting planes proposed, Split cuts (Balas [5]) which are equivalent to the Gomory Mixed Integer cuts (Gomory [21]) and the Mixed Integer Rounding inequalities (Nemhauser and Wolsey [30]) form one of the most successful classes of cutting planes used to solve
Split Rank of Triangle and Quadrilateral Inequalities
, 2009
"... A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and nonnegative continuous variables. Recently Andersen et al. [3] and Cornuéjols and Margot [17] showed that the facetdefining inequalities of this set are eithe ..."
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Cited by 17 (1 self)
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A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and nonnegative continuous variables. Recently Andersen et al. [3] and Cornuéjols and Margot [17] showed that the facetdefining inequalities of this set are either split cuts or intersection cuts obtained from latticefree triangles and quadrilaterals. Through a result by Cook et al. [15], it is known that one particular class of facetdefining triangle inequality does not have a finite split rank. In this paper, we show that all other facetdefining triangle and quadrilateral inequalities have a finite splitrank. The proof is constructive and given a facetdefining triangle or quadrilateral inequality we present an explicit sequence of split inequalities that can be used to generate it.
Lifting group inequalities and an application to mixing inequalities
, 2009
"... Given a valid inequality for the mixed integer infinite group relaxation, a lifting based approach is presented that can be used to strengthen this inequality. Bounds on the solution of the corresponding lifting problem and some necessary conditions for the lifted inequality to be minimal for the mi ..."
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Cited by 13 (0 self)
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Given a valid inequality for the mixed integer infinite group relaxation, a lifting based approach is presented that can be used to strengthen this inequality. Bounds on the solution of the corresponding lifting problem and some necessary conditions for the lifted inequality to be minimal for the mixed integer infinite group relaxation are presented. Finally, these results are applied to generate a strengthened version of the mixing inequality that provides a new class of extreme inequalities for the tworow mixed integer infinite group relaxation.
Network formulations of mixedinteger programs
 In preparation
, 2006
"... We consider mixedinteger sets of the type MIX TU = {x: Ax ≥ b; xi integer, i ∈ I}, where A is a totally unimodular matrix, b is an arbitrary vector and I is a nonempty subset of the column indices of A. We show that the problem of checking nonemptiness of a set MIX TU is NPcomplete even in the cas ..."
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Cited by 11 (6 self)
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We consider mixedinteger sets of the type MIX TU = {x: Ax ≥ b; xi integer, i ∈ I}, where A is a totally unimodular matrix, b is an arbitrary vector and I is a nonempty subset of the column indices of A. We show that the problem of checking nonemptiness of a set MIX TU is NPcomplete even in the case in which the system describes mixedinteger network flows with halfintegral requirements on the nodes. This is in contrast to the case where A is totally unimodular and contains at most two nonzeros per row. Denoting such mixedinteger sets by MIX 2TU, we provide an extended formulation for the convex hull of MIX 2TU whose constraint matrix is a dual network matrix with an integral righthandside vector. The size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of conv(MIX 2TU). When this number is polynomial in the dimension of the matrix A, the extended formulation is of polynomial size. If, in addition, the corresponding list of fractional parts can be computed efficiently, then our result provides a polynomial algorithm for the optimization problem over MIX 2TU. We show that there are instances for which this list is of exponential size, and we also give conditions under which it is short and can be efficiently computed. Finally we show that these results for the set MIX 2TU provide a unified framework leading to polynomialsize extended formulations for several generalizations of mixing sets and lotsizing sets studied in the last few years.