The Continuous Mixing Polyhedron

Vyve, Mathieu Van

doi:10.1287/moor.1040.0130

Cited by 45 publications

(29 citation statements)

References 13 publications

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“…It follows from their work that the internal description of conv(X CM IX ) has exponential size. Van Vyve [48] has provided a new more compact extended formulation that only involves O(n) additional variables, and has shown that the separation problem in the original space can be solved by flow techniques.…”

Section: Continuous Mixing Setmentioning

confidence: 99%

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Extended formulations in combinatorial optimization

2010

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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, Minkowski-Weyl'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 Faenza-Kaibel extended formulation for orbitopes.

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Section: Continuous Mixing Setmentioning

confidence: 99%

“…That is, continuous variables are expressed as a combination of few auxiliary integer variables, thus reducing the original problem to a pure integer one. This approach has been particularly successful in several problems arising in lot-sizing [42], [30], [40], [48], [14] (see [43] for a survey).…”

Section: Variable Discretizationmentioning

confidence: 99%

Extended formulations in combinatorial optimization

2010

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“…Let Q be the polyhedron on the space of the variables {(x i , µ i ℓ ), i ∈ N, ℓ ∈ K ∪{0}} defined by the inequalities (10)-(15) corresponding to inequalities (16), (17) Proof: Theorem 5 shows that every minimal face of Q contains a vector (x,μ) with integral µ. So the same holds for Q I , which is a face of Q.…”

Section: Ray Of the Polyhedron Which Is The Convex Hull Of The Vectormentioning

confidence: 95%

“…An extended formulation for conv(CM IX) which is compact was given by Miller and Wolsey [13]. Later Van Vyve [16] gave a more compact extended formulation and a linear inequality description of conv(CM IX) in the original space.…”

Section: The Continuous Mixing Setmentioning

confidence: 99%

“…More specifically the motivation for this research was in part to understand the full scope of the proof technique based on fractionalities used in studying the mixing set and its generalizations [2,3,5,18,9,13,16], and also to see whether TU matrices play as important a role in mixed-integer programming as they do in pure integer programming.…”

Section: Introductionmentioning

confidence: 99%

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Network Formulations of Mixed-Integer Programs

et al. 2006

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We consider mixed-integer sets of the type M IX T U = {x : Ax ≥ b; x i 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 M IX T U is NP-complete even in the case in which the system describes mixed-integer network flows with half-integral 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 mixed-integer sets by M IX 2T U , we provide an extended formulation for the convex hull of M IX 2T U whose constraint matrix is a dual network matrix with an integral right-hand-side 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(M IX 2T U ). 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 M IX 2T U . 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 M IX 2T U provide a unified framework leading to polynomial-size extended formulations for several generalizations of mixing sets and lot-sizing sets studied in the last few years.

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Mixing Sets

Wolsey

2011

Wiley Encyclopedia of Operations Research and Management Science

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A mixing set is a simple mixed integer set that has a rich structure and has turned out to provide very effective relaxations for a variety of lot‐sizing problems. Here we describe some applications, some of the basic results on valid inequalities and extended formulations. We then present several of the generalizations that have been studied, and indicate the major results obtained for these general mixing sets.

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The Continuous Mixing Polyhedron

Cited by 45 publications

References 13 publications

Extended formulations in combinatorial optimization

Extended formulations in combinatorial optimization

Network Formulations of Mixed-Integer Programs

Mixing Sets

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