## Corner Polyhedron and Intersection Cuts (2010)

Citations: | 5 - 2 self |

### BibTeX

@MISC{Conforti10cornerpolyhedron,

author = {Michele Conforti and Gérard Cornuéjols and Giacomo Zambelli},

title = {Corner Polyhedron and Intersection Cuts},

year = {2010}

}

### OpenURL

### Abstract

A recent paper of Andersen, Louveaux, Weismantel and Wolsey has generated a renewed interest in Gomory’s corner polyhedron and Balas ’ intersection cuts. We survey these two approaches and the recent developments in multi-row cuts. 1

### Citations

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- Rockafellar
- 1970
(Show Context)
Citation Context ...r. Note that the intersection cut defined by T is ∑ 3.2 The gauge function j∈N j∈N γjxj ≥ 1. Let K ⊆ R n be a closed, convex set with the origin in its interior. A standard concept in convex analysis =-=[64, 73]-=- is that of gauge (sometimes called Minkowski function), which is the function γK defined by γK(r) = inf{t > 0 : r t ∈ K}, for all r ∈ Rn . It is the smallest scalar t > 0 such that r t belongs to K. ... |

421 |
Lectures on functional equations and their applications
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(Show Context)
Citation Context ... 1 − f 1 Figure 6: Examples of minimal valid functions (q = 1) Example 6.5. Figure 6 gives examples of minimal valid functions for model (19) with q = 1. The functions are represented in the interval =-=[0, 1]-=- and are defined elsewhere by periodicity. Note the symmetry relative to the points ( 1−f 1 f 1 2 , 2 ) and (1 − 2 , 2 ). Checking subadditivity is a nontrivial task. Gomory, Johnson and Evans [60] sh... |

234 |
An algorithm for integer solutions to linear programs
- Gomory
- 1963
(Show Context)
Citation Context ...y and Johnson on the corner polyhedron [55, 57], dating back to the early 1970s. Split cuts are a classical example of intersection cuts. They are equivalent [72] to Gomory’s mixed integer (GMI) cuts =-=[53]-=-, which are generated from a single equation valid for the linear relaxation of a mixed integer set. Most cutting planes currently implemented in software are split cuts, such as GMI cuts from tableau... |

225 |
A lift-and-project cutting plane algorithm for mixed 0–1 programs
- Balas, Ceria, et al.
- 1993
(Show Context)
Citation Context ...st cutting planes currently implemented in software are split cuts, such as GMI cuts from tableau rows, mixed integer rounding inequalities [68], lifted cover inequalities [11], lift-and-project cuts =-=[12]-=-. A flurry of current research investigates new families of intersection cuts derived from multiple rows of the tableau, and that are not split cuts. This survey covers both classical and recent resul... |

128 |
Fundamentals of Convex Analysis
- Hirriart-Urruty, Lemaréchal
- 2001
(Show Context)
Citation Context ...r. Note that the intersection cut defined by T is ∑ 3.2 The gauge function j∈N j∈N γjxj ≥ 1. Let K ⊆ R n be a closed, convex set with the origin in its interior. A standard concept in convex analysis =-=[64, 73]-=- is that of gauge (sometimes called Minkowski function), which is the function γK defined by γK(r) = inf{t > 0 : r t ∈ K}, for all r ∈ Rn . It is the smallest scalar t > 0 such that r t belongs to K. ... |

76 |
Facets of the knapsack polytope
- Balas
- 1975
(Show Context)
Citation Context ...n of a mixed integer set. Most cutting planes currently implemented in software are split cuts, such as GMI cuts from tableau rows, mixed integer rounding inequalities [68], lifted cover inequalities =-=[11]-=-, lift-and-project cuts [12]. A flurry of current research investigates new families of intersection cuts derived from multiple rows of the tableau, and that are not split cuts. This survey covers bot... |

76 |
Some polyhedra related to combinatorial problems
- Gomory
- 1969
(Show Context)
Citation Context ..., besides nonnegativity constraints, the facet defining inequalities can be derived from splits, triangles and quadrilaterals. This elegant result has sparked a renewed interest in the work of Gomory =-=[60]-=- and Gomory and Johnson [62] on the corner polyhedron, and of Balas on intersection cuts generated from convex sets [10], dating back to the early 1970s. Split cuts are a classical example of intersec... |

67 |
Intersection cuts — a new type of cutting planes for integer programming
- Balas
- 1971
(Show Context)
Citation Context ...defining inequalities can be derived from splits, triangles and quadrilaterals. This elegant result has sparked a renewed interest in the work of Balas on intersection cuts generated from convex sets =-=[9]-=-, and of Gomory and Johnson on the corner polyhedron [55, 57], dating back to the early 1970s. Split cuts are a classical example of intersection cuts. They are equivalent [72] to Gomory’s mixed integ... |

67 | Aggregation and mixed integer rounding to solve MIPs
- Marchand, Wolsey
- 2001
(Show Context)
Citation Context ...I) cuts [58], which are generated from a single equation. Most cutting planes currently implemented in software are split cuts, such as GMI cuts from tableau rows, mixed integer rounding inequalities =-=[73]-=- and lift-and-project cuts [12]. A flurry of current research investigates intersection cuts derived from multiple rows of the tableau. 1 Dipartimento di Matematica Pura e Applicata, Università di Pad... |

67 |
A recursive procedure for generating all cuts for 0-1 mixed integer programs
- Nemhauser, Wolsey
- 1990
(Show Context)
Citation Context ... the corner polyhedron, and of Balas on intersection cuts generated from convex sets [10], dating back to the early 1970s. Split cuts are a classical example of intersection cuts. They are equivalent =-=[77]-=- to Gomory’s mixed integer (GMI) cuts [58], which are generated from a single equation. Most cutting planes currently implemented in software are split cuts, such as GMI cuts from tableau rows, mixed ... |

34 |
Cutting Planes from Two Rows of a Simplex Tableau
- Andersen, Louveaux, et al.
- 2007
(Show Context)
Citation Context ...generated a renewed interest in Gomory’s corner polyhedron and Balas’ intersection cuts. We survey these two approaches and the recent developments in multi-row cuts. 1 Introduction In a recent paper =-=[6]-=-, Andersen, Louveaux, Weismantel and Wolsey study a mixed integer linear programming (MILP) in tableau form, where the basic variables are free integer variables and the nonbasic variables are continu... |

33 |
Strengthening cuts for mixed integer programs
- Balas, Jeroslow
- 1980
(Show Context)
Citation Context ...d function for Mf . Proof. Given a minimal valid function ψ for Rf , (ψ, ψ) is a valid function for Mf . Apply Lemma 8.5 to (ψ, ψ). The function π ′ defined in Corollary 8.6 is a trivial lifting of ψ =-=[13, 57]-=-. Let (ψ, π) be a minimal valid function for Mf where in addition ψ is a minimal function for Rf . We exhibit and example where this latter property completely determines the function π. Example 8.7. ... |

32 | Optimizing over the split closure
- Balas, Saxena
- 2008
(Show Context)
Citation Context ...Campelo, Conforti, Cornuéjols and Zambelli [20]. Dey and Wolsey [49] combine the trivial lifting approach described above with traditional sequential lifting. 9 Computations Computational experiments =-=[14, 16, 19, 40, 42, 51]-=- help us assess the value of intersection cuts in practice, particularly those defined by splits but also those derived from multiple rows. Balas and Saxena [16] and Dash, Günlük and Lodi [40] solved ... |

30 |
On the facets of mixed integer programs with two integer variables and two constraints
- Cornuéjols, Margot
(Show Context)
Citation Context ...rsen, Louveaux, Weismantel and Wolsey [6] who proved that, when |B| = p = 2, the intersection cuts defined by splits, triangles and quadrilaterals describe corner(B) completely. Cornuéjols and Margot =-=[35]-=- characterize exactly which splits, triangles and quadrilaterals produce intersection cuts that are facets of corner(B), again in the case when |B| = p = 2. 12f f f Figure 5: Maximal lattice-free con... |

26 |
A Course in Convexity, Graduate
- Barvinok
- 2002
(Show Context)
Citation Context ...d positively homogeneous, then g(tx) = 0 for all t > 0. Hence g(x) = 0 = inf{t > 0 : x t ∈ K}. 4 Maximal lattice-free convex sets For a good reference on lattices and convexity, we recommend Barvinok =-=[17]-=-. Here we will only work with the integer lattice Z p . By Remark 3.2, the best possible intersection cuts are the ones defined by full-dimensional maximal Z p × R n−p -free convex sets in R n , that ... |

26 | Lifting integer variables in minimal inequalities corresponding to lattice-free triangles
- Dey, Wolsey
(Show Context)
Citation Context ...a minimal valid function for Mf where in addition ψ is a minimal function for Rf . We exhibit and example where this latter property completely determines the function π. Example 8.7. (Dey and Wolsey =-=[46, 47]-=-) Let q = 2. Consider the maximal lattice-free triangle K = conv( ( ) ( ) ( ) 0 2 0 0 , 0 , 2 ), assume that f is a point in the interior of K (see Figure 8) and let ψK be the gauge of K − f. By Theor... |

25 | Minimal Valid Inequalities for Integer Constraints, technical report
- Borozan, Cornuejols
- 2007
(Show Context)
Citation Context ...inuous Infinite Relaxation We consider the following model, where all nonbasic variables yj, j ∈ N are continuous. fi + ∑ j∈N rj i yj ∈ Z for i = 1, . . . , q yj ≥ 0 for j ∈ N. Borozan and Cornuéjols =-=[29]-=- and Basu, Conforti, Cornuéjols and Zambelli [22] studied the continuous infinite relaxation, obtained from (22) by augmenting the space of variables yj, j ∈ N, to an infinite-dimensional space {yr, r... |

21 |
On the relation between integer and noninteger solutions to linear programs
- Gomory
- 1965
(Show Context)
Citation Context ... are violated by the point ¯x. Typically, ¯x is an optimal solution of the LP relaxation of an MILP having (1) as feasible set. The key idea is to work with the corner polyhedron introduced by Gomory =-=[54, 55]-=-, which is obtained from (1) by dropping the nonnegativity restriction on all the basic variables xi, i ∈ B. Note that in this relaxation we can drop the constraints xi = ¯bi − ∑ j∈N āijxj for all i ∈... |

21 | A note on the split rank of intersection cuts - DEY |

20 |
Corner polyhedra and their connection with cutting planes, Mathematical Programming 96
- Gomory, Johnson, et al.
- 2003
(Show Context)
Citation Context ...er variables, (3) can be reformulated as follows ∑ j∈N āijxj ≡ ¯bi mod 1 for i ∈ B xj ∈ Z for j ∈ {1, . . . , p} ∩ N xj ≥ 0 for j ∈ N. This point of view was extensively studied by Gomory and Johnson =-=[55, 56, 57, 58, 59, 60, 61]-=-. We will come back to it in Section 6. (7) 3 Intersection cuts We describe a paradigm introduced by Balas [9] for constructing inequalities that are valid for the corner polyhedron and that cut off t... |

20 | Constrained infinite group relaxations of MIPs
- DEY, WOLSEY
(Show Context)
Citation Context ...spondence between extreme inequalities for the infinite model (23) and extreme inequalities for the finite problem (22). This theorem appears in [20] and is very similar to a result of Dey and Wolsey =-=[52]-=-. Theorem 6.5. Let B be a maximal lattice-free convex set in Rq with f in its interior. Let L = lin(B) and let P = B ∩ (f + L⊥ ). Then B = P + L, L is a rational space, and P is a polytope. Let v1 , .... |

16 |
An observation on the structure of production sets with indivisibilities
- Scarf
- 1977
(Show Context)
Citation Context ...d with the inequality αtx ≤ αtz. By construction, K ′′ does not contain any point of Zp in its interior and properly contains K, contradicting the maximality of K. ✷ Doignon [50], Bell [27] and Scarf =-=[74]-=- show the following. Theorem 4.3. Any full-dimensional maximal lattice-free convex set K ⊆ R p has at most 2 p facets. Proof. By Theorem 4.2, each facet F contains an integral point x F in its relativ... |

15 |
A theorem concerning the integer lattice
- Bell
- 1977
(Show Context)
Citation Context ...been substituted with the inequality αtx ≤ αtz. By construction, K ′′ does not contain any point of Zp in its interior and properly contains K, contradicting the maximality of K. ✷ Doignon [50], Bell =-=[27]-=- and Scarf [74] show the following. Theorem 4.3. Any full-dimensional maximal lattice-free convex set K ⊆ R p has at most 2 p facets. Proof. By Theorem 4.2, each facet F contains an integral point x F... |

14 | Split rank of triangle and quadrilateral inequalities
- Dey, Louveaux
(Show Context)
Citation Context .... Adding this single inequality to the formulation of P , we obtain conv(S). Yet y ≤ 0 does not have finite split rank [33]. This example has been generalized by Li and Richard [66]. Dey and Louveaux =-=[43]-=- study the split rank of intersection cuts for problems with two integer variables (model (3) where |B| = p = 2). Surprisingly, they show that all intersection cuts have finite split rank except for t... |

13 | On the relative strength of split, triangle and quadrilateral cuts. To Appear
- Basu, Bonami, et al.
- 2009
(Show Context)
Citation Context ...xists a (possibly infeasible) basis B of Ax = b such that αx ≤ β is an intersection cut of corner(B) defined by a split. 145.2 Triangle and quadrilateral closures Basu, Bonami, Cornuéjols and Margot =-=[18]-=- consider model (3) when |B| = p = 2. As before, P (B) denotes the LP relaxation of (3). Let S ⊂ R n denote the split closure of P (B). Define the triangle closure T of P (B) to be the subset of R n s... |

12 | Minimal inequalities for an infinite relaxation of integer programs
- Basu, Conforti, et al.
- 2010
(Show Context)
Citation Context ...r with Theorem 7.5 justifies the choice that we made in this section to define ψ with its values in R. Model (23) was extended to the case where f + ∑ r∈R q ryr ∈ Z q ∩ P for some rational polyhedron =-=[10, 23, 26, 48, 52, 62]-=-. We may assume that S = Z q ∩ P is full-dimensional. In this model, minimal valid inequalities are still of the form (24) but now they may have negative coefficients ψ(r). Despite this, many of the k... |

12 | A geometric perspective on lifting
- Conforti, Cornuéjols, et al.
- 2011
(Show Context)
Citation Context ..., since, for any d ∈ R2 , it gives a construction for an integral vector wd such that d + wd ∈ RK. This geometric perspective on lifting was extended to general q by Conforti, Cornuéjols and Zambelli =-=[32]-=- and Basu, Campelo, Conforti, Cornuéjols and Zambelli [20]. Dey and Wolsey [49] combine the trivial lifting approach described above with traditional sequential lifting. 9 Computations Computational e... |

12 |
A Recursive Procedure to Generate All Cuts for 0-1
- Nemhauser, Wolsey
- 1990
(Show Context)
Citation Context ...nerated from convex sets [9], and of Gomory and Johnson on the corner polyhedron [55, 57], dating back to the early 1970s. Split cuts are a classical example of intersection cuts. They are equivalent =-=[72]-=- to Gomory’s mixed integer (GMI) cuts [53], which are generated from a single equation valid for the linear relaxation of a mixed integer set. Most cutting planes currently implemented in software are... |

11 |
Integer Programming and Convex Analysis: Intersection Cuts from Outer
- Balas
- 1972
(Show Context)
Citation Context ...r with Theorem 7.5 justifies the choice that we made in this section to define ψ with its values in R. Model (23) was extended to the case where f + ∑ r∈R q ryr ∈ Z q ∩ P for some rational polyhedron =-=[10, 23, 26, 48, 52, 62]-=-. We may assume that S = Z q ∩ P is full-dimensional. In this model, minimal valid inequalities are still of the form (24) but now they may have negative coefficients ψ(r). Despite this, many of the k... |

11 |
Some Continuous Functions Related to
- Gomory, Johnson
- 1972
(Show Context)
Citation Context ...les and quadrilaterals. This elegant result has sparked a renewed interest in the work of Balas on intersection cuts generated from convex sets [9], and of Gomory and Johnson on the corner polyhedron =-=[55, 57]-=-, dating back to the early 1970s. Split cuts are a classical example of intersection cuts. They are equivalent [72] to Gomory’s mixed integer (GMI) cuts [53], which are generated from a single equatio... |

11 | On Degenerate Multi-Row Gomory Cuts
- Zambelli
- 2009
(Show Context)
Citation Context ...∪{+∞} instead of just R. Indeed, valid inequalities of this type exist for (23). For this reason, Borozan and Cornuéjols [29] consider valid functions ψ that take values in R ∪ {+∞}. However Zambelli =-=[76]-=- showed that the extension to R ∪ {+∞} is never needed for the finite model (22), in the sense that the coefficients of every valid inequality for (22) are always defined by some finite valid function... |

10 |
Experiments with two row tableau cuts
- Dey, Lodi, et al.
- 2010
(Show Context)
Citation Context ...Campelo, Conforti, Cornuéjols and Zambelli [20]. Dey and Wolsey [49] combine the trivial lifting approach described above with traditional sequential lifting. 9 Computations Computational experiments =-=[14, 16, 19, 40, 42, 51]-=- help us assess the value of intersection cuts in practice, particularly those defined by splits but also those derived from multiple rows. Balas and Saxena [16] and Dash, Günlük and Lodi [40] solved ... |

9 |
Some Polyhedra Related to
- Gomory
- 1969
(Show Context)
Citation Context ...les and quadrilaterals. This elegant result has sparked a renewed interest in the work of Balas on intersection cuts generated from convex sets [9], and of Gomory and Johnson on the corner polyhedron =-=[55, 57]-=-, dating back to the early 1970s. Split cuts are a classical example of intersection cuts. They are equivalent [72] to Gomory’s mixed integer (GMI) cuts [53], which are generated from a single equatio... |

8 |
Convexity in Cristallographical Lattices
- Doignon
- 1973
(Show Context)
Citation Context ...t that has been substituted with the inequality αtx ≤ αtz. By construction, K ′′ does not contain any point of Zp in its interior and properly contains K, contradicting the maximality of K. ✷ Doignon =-=[50]-=-, Bell [27] and Scarf [74] show the following. Theorem 4.3. Any full-dimensional maximal lattice-free convex set K ⊆ R p has at most 2 p facets. Proof. By Theorem 4.2, each facet F contains an integra... |

8 |
Characterization of facets for multiple right-hand side choice linear programs
- Johnson
- 1981
(Show Context)
Citation Context ...r with Theorem 7.5 justifies the choice that we made in this section to define ψ with its values in R. Model (23) was extended to the case where f + ∑ r∈R q ryr ∈ Z q ∩ P for some rational polyhedron =-=[10, 23, 26, 48, 52, 62]-=-. We may assume that S = Z q ∩ P is full-dimensional. In this model, minimal valid inequalities are still of the form (24) but now they may have negative coefficients ψ(r). Despite this, many of the k... |

7 | A Counterexample to a Conjecture of Gomory and Johnson, manuscript
- Basu, Conforti, et al.
- 2008
(Show Context)
Citation Context ... < 1 1 0 1 − f 1 Figure 7: A discontinuous minimal valid function Gomory and Johnson [59] conjectured that extreme valid functions are always piecewise linear. Basu, Conforti, Cornuéjols and Zambelli =-=[21]-=- disprove this conjecture. However Corollary 4.5 shows that the Gomory-Johnson conjecture is ”almost” true, and that pathologies only arise when we consider an infinite number of integer variables. 6.... |

7 |
Constrained Infinite Group Relaxations of MIPs, manuscript
- Dey, Wolsey
- 2009
(Show Context)
Citation Context ...spondence between extreme inequalities for the infinite model (23) and extreme inequalities for the finite problem (22). This theorem appears in [22] and is very similar to a result of Dey and Wolsey =-=[48]-=-. Theorem 7.5. Let B be a maximal lattice-free convex set in Rq with f in its interior. Let L = lin(B) and let P = B ∩ (f + L⊥ ). Then B = P + L, L is a rational space, and P is a polytope. Let v1 , .... |

7 |
Generalized Mixed Integer Rounding Inequalities: Facets for Infinite
- Kianfar, Fathi
- 2009
(Show Context)
Citation Context ...es of known valid inequalities satisfy the two-slope theorem, such as GMI cuts, the 2-step MIR inequality of Dash and Günlük [39] and, more generally, the n-step MIR inequalities of Kianfar and Fathi =-=[65]-=-. 7 Continuous Infinite Relaxation We consider the following model, where all nonbasic variables yj, j ∈ N are continuous. fi + ∑ j∈N rj i yj ∈ Z for i = 1, . . . , q yj ≥ 0 for j ∈ N. Borozan and Cor... |

6 |
On Lifting Integer Variables
- Basu, Campelo, et al.
(Show Context)
Citation Context ...ty follows from the fact that (x, y) ∈ Mf and that (π, ψ) is a valid function for Mf , the second inequality follows because, by Lemma 8.1 π(r ∗ ) ≤ ψ(r ∗ ). Now (29) implies π(r ∗ ) = ψ(r ∗ ). 29In =-=[20]-=- it is proven that if r ∗ does not satisfy Property (29), then π ′ (r ∗ ) < ψ(r ∗ ) for some minimal valid function (π ′ , ψ). Lemma 8.5. Let (π, ψ) be a valid function for Mf and let π ′(r) = infw∈Z ... |

6 |
On the Existence of Optimal Solutions to Integer and
- Meyer
- 1974
(Show Context)
Citation Context ... x ′′ i = ¯bi − ∑ j∈N j , x′′ belongs to corner(B). This shows that corner(B) is nonempty. Since P (B) is a rational polyhedron, the recession cones of P (B) and corner(B) coincide by Meyer’s theorem =-=[69]-=-. Since the dimension of both P (B) and its recession cone is |N| and corner(B) ⊆ P (B), the dimension of corner(B) is |N|. Example 2.3. Consider the pure integer program max 1 2 x2 + x3 x1 + x2 + x3 ... |

5 | On the Rank of Mixed 0,1 - Cornuéjols, Li - 2002 |

5 |
Thoughts about Integer Programming, 50th Anniversary Symposium of OR
- Gomory
(Show Context)
Citation Context ...er variables, (3) can be reformulated as follows ∑ j∈N āijxj ≡ ¯bi mod 1 for i ∈ B xj ∈ Z for j ∈ {1, . . . , p} ∩ N xj ≥ 0 for j ∈ N. This point of view was extensively studied by Gomory and Johnson =-=[55, 56, 57, 58, 59, 60, 61]-=-. We will come back to it in Section 6. (7) 3 Intersection cuts We describe a paradigm introduced by Balas [9] for constructing inequalities that are valid for the corner polyhedron and that cut off t... |

5 |
Composite Lifting of Group Inequalities and an Application to Two-Row Mixing Inequalities, Discrete Optimization 7
- Dey, Wolsey
- 2010
(Show Context)
Citation Context ... perspective on lifting was extended to general q by Conforti, Cornuéjols and Zambelli [31], Basu, Campelo, Conforti, Cornuéjols and Zambelli [18], and Basu, Cornuéjols and Köppe [22]. Dey and Wolsey =-=[53]-=- combine the trivial lifting approach described above with traditional sequential lifting. Dey and Richard [48] give facet defining inequalities for the infinite relaxation. Richard and Dey [78] give ... |

5 |
Strengthening lattice-free cuts using non-negativity. Available on OptimizationOnline
- Fukasawa, Günlük
- 2009
(Show Context)
Citation Context ...r with Theorem 6.5 justifies the choice that we made in this section to define ψ with its values in R. Model (23) was extended to the case where f + ∑ r∈R q ryr ∈ Z q ∩ P for some rational polyhedron =-=[11, 21, 25, 52, 57, 67]-=-. In this model, minimal valid inequalities are still of the form (24) but now they may have negative coefficients ψ(r). Many of the key results still hold in this more general model. In particular it... |

4 | Intersection Cuts with Infinite Split Rank
- Basu, Cornuéjols, et al.
- 2010
(Show Context)
Citation Context ...pt for the ones defined by lattice-free triangles with integral 16vertices and an integral point in the middle of each side. The triangle K defined above is of this type. Basu, Cornuéjols and Margot =-=[24]-=- extend this result to more than two integer variables. To state their theorem, we first need to define the 2-hyperplane property. A set S of points in R p is 2-partitionable if either |S| ≤ 1 or ther... |

4 | Convex Sets and Minimal Sublinear Functions
- Basu, Cornuéjols, et al.
(Show Context)
Citation Context |

4 | Two Dimensional Lattice-free Cuts and Asymmetric Disjunctions for Mixed-integer Polyhedra
- Dash, Dey, et al.
- 2010
(Show Context)
Citation Context ...s defined by triangles. Using a different probabilistic approach, Basu, Cornuéjols and Molinaro [25] show that the split closure is a good approximation of corner(B) on average. Dash, Dey and Günlük, =-=[38]-=- proved an intriguing result for model (3) when |B| = p = 2. They showed that corner(B) is defined entirely by disjunctive cuts from crooked cross disjunctions of the form {x ∈ R 2 : π 1 x ≤ π 1 0 , (... |

4 |
On the Group Problem for Mixed Integer
- Johnson
- 1974
(Show Context)
Citation Context ...er variables, (3) can be reformulated as follows ∑ j∈N āijxj ≡ ¯bi mod 1 for i ∈ B xj ∈ Z for j ∈ {1, . . . , p} ∩ N xj ≥ 0 for j ∈ N. This point of view was extensively studied by Gomory and Johnson =-=[55, 56, 57, 58, 59, 60, 61]-=-. We will come back to it in Section 6. (7) 3 Intersection cuts We describe a paradigm introduced by Balas [9] for constructing inequalities that are valid for the corner polyhedron and that cut off t... |

4 |
Maximal Lattice-Free Polyhedra: Finiteness and an Explicit Description in Dimension Three
- Averkov, Wagner, et al.
- 2010
(Show Context)
Citation Context ...al vectors. Del Pia and Weismantel [43] show that the convex hull of a mixed integer set can be obtained with inequalities derived from integral lattice-free polyhedra. Averkov, Wagner and Weismantel =-=[9]-=- show that that in fixed dimension, up to unimodular transformations, there exist a finite number of maximal polyhedra (with respect to inclusion), among the integral lattice-free polyhedra. Propertie... |

4 |
A Heuristic to Generate Rank-1
- Dash, Goycoolea
(Show Context)
Citation Context ...h variance. Other split cuts, such as MIR, lift-and-project, reduce-and-split, typically improve the gap closed to 40% or more. We just mention a few recent studies along these lines here [27], [36], =-=[37]-=-. Some initial results have been obtained on intersection cuts from multiple rows ( Basu, Bonami, Cornuéjols and Margot [17], Dey, Lodi, Tramontani and Wolsey [45], Espinoza [55]). 32The work of Dey,... |