### Citations

1415 |
Integer and Combinatorial Optimization
- Nemhauser, Wolsey
- 1988
(Show Context)
Citation Context ...ngian subproblem) for a given sequence of dual multipliers u. This is an optimization problem over P ′ and can be solved effectively by assumption. Both of these approaches are described in detail in =-=[29]-=-. 3.2 Dantzig-Wolfe Decomposition The approach of Dantzig-Wolfe decomposition (DWD) [8] is to reformulate (IP) by implicitly requiring the solution to be a member of F ′, while explicitly enforcing th... |

387 |
The Decomposition Principle for Linear Programs.”
- Dantzig, Wolfe
- 1960
(Show Context)
Citation Context ...oblem over P ′ and can be solved effectively by assumption. Both of these approaches are described in detail in [29]. 3.2 Dantzig-Wolfe Decomposition The approach of Dantzig-Wolfe decomposition (DWD) =-=[8]-=- is to reformulate (IP) by implicitly requiring the solution to be a member of F ′, while explicitly enforcing the inequalities [A′′, b ′′]. Relaxing the integrality constraints on the variables from ... |

360 |
The lagrangian relaxation method for solving integer programming problems.
- Fisher
- 2004
(Show Context)
Citation Context ... principle of decomposition. In Section 4, we show that these methods are very closely related. 3.1 Lagrangian Relaxation For a given vector of dual multipliers u ∈ Rm′′ + , the Lagrangian relaxation =-=[9, 6, 28, 23]-=- of (IP) is given by time. zLR(u) = min s∈F ′{(c⊤ − u ⊤ A ′′ )s + u ⊤ b ′′ }. (LR) 1 The term practical is not defined rigorously, but denotes an algorithm with a “reasonable” average-case running 2I... |

295 |
An algorithm for integer solutions to linear programs,"
- Gomory
- 1963
(Show Context)
Citation Context ...Using this mapping, we can see that zDW = c ⊤ ˆxDW . Since ˆxDW must lie within P ′ ⊆ Q ′ and also within Q ′′ , this shows that zDW ≥ zLP . 3.3 Cutting Plane Method Although the cutting plane method =-=[16, 7, 30, 20]-=- is not usually thought of as a decomposition method, it can in fact be viewed as such. Under our assumption that the separation problem associated with P ′ can be solved effectively, a cutting plane ... |

223 |
The Travelling Salesman Problem and Minimum Spanning Trees: Part II.”
- Held, Carp
- 1971
(Show Context)
Citation Context ... the problems for which decomposition approaches have been proposed in the literature includes the Multicommodity Flow Problem [11], the Cutting Stock Problem [15, 37], the Traveling Salesman Problem =-=[19]-=-, the Generalized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Colo... |

192 | Lagrangean relaxation for integer programming
- GEOFFRION
- 1974
(Show Context)
Citation Context ...ationship of the Techniques The following well-known result showed that the three approaches just described are simply three different algorithms for computing the same quantity. Theorem 1 (Geoffrion =-=[14]-=-) zIP ≥ c ⊤ ˆxDW = zLD = zDW = zCP = min{c ⊤ x | P ′ ∩ Q ′′ } ≥ zLP . We refer to the quantity zD = max{c ⊤ x | P ′ ∩ Q ′′ } as the decomposition bound. The proof of this result follows directly from ... |

173 |
A linear programming approach to the cutting stock problem.
- Gilmore, Gomory
- 1963
(Show Context)
Citation Context ...combinatorial problems. A small sample of the problems for which decomposition approaches have been proposed in the literature includes the Multicommodity Flow Problem [11], the Cutting Stock Problem =-=[15, 37]-=-, the Traveling Salesman Problem [19], the Generalized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Sc... |

144 | Selected topics in column generation,
- Lubbecke, Desrosiers
- 2002
(Show Context)
Citation Context ...LR). Because the DWLP is solved by column generation, methods that use a DWLP to obtain bounds within a branch-bound-bound algorithm are sometimes referred to generically as column generation methods =-=[5, 24]-=-. In fact, most column generation algorithms for solving integer programs, even those for which the column generation derives directly from a “natural” formulation, can be seen as arising from the app... |

141 |
The cutting plane method for solving convex programs
- Kelley
- 1960
(Show Context)
Citation Context ...u ⊤ A ′′ )s ∀s ∈ F ′ }. (LDLP) Of course, this linear program has a large number of constraints and so must be solved by means of a cut generation algorithm (known as Kelley’s cutting plane algorithm =-=[21]-=-). In either approach to solving the Lagrangian dual, most of the computational effort goes into evaluating zLR(u) (called the Lagrangian subproblem) for a given sequence of dual multipliers u. This i... |

105 |
Discrete Optimization,
- Parker, Rardin
- 1988
(Show Context)
Citation Context ...ion is used to obtain a bound in each node are known as branch-and-price algorithms [34, 35, 40]. It is easy to verify that (DWLP) is an LP dual of (LDLP), which immediately shows that zDW = zLD (see =-=[31]-=- for a detailed treatment of this fact). Hence, zDW is a valid lower bound on zIP that we s∈F ′ 2 Note that we can equivalently replace F ′ with the extreme points of P ′ . s∈F ′ 3call the DW bound. ... |

93 | A column generation approach for graph coloring.
- Mehrotra, Trick
- 1996
(Show Context)
Citation Context ...ized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem =-=[27]-=-, and the Capacitated Vehicle Routing Problem (CVRP) [1, 10, 32, 12]. To expose the desired substructure, a common approach is to relax a set of “complicating constraints.” This is the approach taken ... |

80 |
A Branch-and-price algorithm for the generalized assignment problem
- SAVELSBERGH
- 1993
(Show Context)
Citation Context ...roaches have been proposed in the literature includes the Multicommodity Flow Problem [11], the Cutting Stock Problem [15, 37], the Traveling Salesman Problem [19], the Generalized Assignment Problem =-=[18, 35]-=-, the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicl... |

77 |
Optimal Solution of Vehicle Routing Problems Using Minimum K-Trees.
- Fisher
- 1994
(Show Context)
Citation Context ... [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicle Routing Problem (CVRP) =-=[1, 10, 32, 12]-=-. To expose the desired substructure, a common approach is to relax a set of “complicating constraints.” This is the approach taken by the Dantzig-Wolfe decomposition, Lagrangian relaxation, and cutti... |

63 |
A suggested computation for maximal multicommodity network flows.
- Ford, Fulkerson
- 1958
(Show Context)
Citation Context ... approaches for many well-known combinatorial problems. A small sample of the problems for which decomposition approaches have been proposed in the literature includes the Multicommodity Flow Problem =-=[11]-=-, the Cutting Stock Problem [15, 37], the Traveling Salesman Problem [19], the Generalized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree... |

63 |
2-path cuts for the vehicle routing problem with time windows.
- Kohl, Desrosiers, et al.
- 1999
(Show Context)
Citation Context ...dure we call price and cut. When employed as the bounding procedure in a branch-and-bound framework, the overall technique is called branch, price, and cut and has been studied by a number of authors =-=[38, 22, 4, 36]-=-. Simultaneous generation of columns and valid inequalities is difficult in general because the addition of valid inequalities may destroy the structure of the column generation subproblem (for a disc... |

61 |
Using Branch-and-Price-and-Cut to Solve Origin-Destination Integer Multicommodity Flow Problems.”
- Barnhart, Hane, et al.
- 1998
(Show Context)
Citation Context ...dure we call price and cut. When employed as the bounding procedure in a branch-and-bound framework, the overall technique is called branch, price, and cut and has been studied by a number of authors =-=[38, 22, 4, 36]-=-. Simultaneous generation of columns and valid inequalities is difficult in general because the addition of valid inequalities may destroy the structure of the column generation subproblem (for a disc... |

60 | An algorithm for the three-index assignment problem.
- Balas, Saltzman
- 1991
(Show Context)
Citation Context ...ating an arbitrary real vector is difficult, but the problem of separating a solution to a given combinatorial relaxation is easy. This notion has been discussed in the literature in several contexts =-=[32, 26, 3]-=- and leads to a separation technique that can be embedded within price and cut[33]. The idea is to replace direct separation of the fractional point (which may be difficult) with separation of members... |

60 | Robust branch-and-cut-and-price for the capacitated vehicle routing problem.
- Fukasawa, Longo, et al.
- 2006
(Show Context)
Citation Context ... [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicle Routing Problem (CVRP) =-=[1, 10, 32, 12]-=-. To expose the desired substructure, a common approach is to relax a set of “complicating constraints.” This is the approach taken by the Dantzig-Wolfe decomposition, Lagrangian relaxation, and cutti... |

57 |
Solution of a large scale traveling salesman problem.
- Dantzig, Fulkerson, et al.
- 1954
(Show Context)
Citation Context ...Using this mapping, we can see that zDW = c ⊤ ˆxDW . Since ˆxDW must lie within P ′ ⊆ Q ′ and also within Q ′′ , this shows that zDW ≥ zLP . 3.3 Cutting Plane Method Although the cutting plane method =-=[16, 7, 30, 20]-=- is not usually thought of as a decomposition method, it can in fact be viewed as such. Under our assumption that the separation problem associated with P ′ can be solved effectively, a cutting plane ... |

56 |
Computational study of a column generation algorithm for bin packing and cutting stock problems.
- Vanderbeck
- 1999
(Show Context)
Citation Context ...literature includes the Multicommodity Flow Problem [11], the Cutting Stock Problem [15, 37], the Traveling Salesman Problem [19], the Generalized Assignment Problem [18, 35], the Bin Packing Problem =-=[39]-=-, the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicle Routing Problem (CVRP) [1, 1... |

54 |
Solving binary cutting stock problems by column generation and branch-and-bound,”
- Vance, Barnhart, et al.
- 1994
(Show Context)
Citation Context ...combinatorial problems. A small sample of the problems for which decomposition approaches have been proposed in the literature includes the Multicommodity Flow Problem [11], the Cutting Stock Problem =-=[15, 37]-=-, the Traveling Salesman Problem [19], the Generalized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Sc... |

46 | On the capacitated vehicle routing problem.
- Ralphs, Kopman, et al.
- 2003
(Show Context)
Citation Context ... [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicle Routing Problem (CVRP) =-=[1, 10, 32, 12]-=-. To expose the desired substructure, a common approach is to relax a set of “complicating constraints.” This is the approach taken by the Dantzig-Wolfe decomposition, Lagrangian relaxation, and cutti... |

46 | Reformulation and decomposition of integer programs, in:
- Vanderbeck, Wolsey
- 2010
(Show Context)
Citation Context ...composition methodologies based on relaxation of constraints and examines how they are used to solve mixed integer linear programs. For a broader overview, including variable restriction methods, see =-=[41]-=-. To simplify the exposition, we consider only pure integer linear programs (ILPs) with finite upper and lower bounds on all variables, so that the set of feasible solutions is finite. The framework c... |

41 | Time-Indexed Formulations for Machine Scheduling Problems: Column Generation.
- Akker, Hurkens, et al.
- 2000
(Show Context)
Citation Context ...alesman Problem [19], the Generalized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem =-=[36]-=-, the Graph Coloring Problem [27], and the Capacitated Vehicle Routing Problem (CVRP) [1, 10, 32, 12]. To expose the desired substructure, a common approach is to relax a set of “complicating constrai... |

39 |
LP-based combinatorial problem solving
- HOFFMAN, PADBERG
- 1985
(Show Context)
Citation Context ...Using this mapping, we can see that zDW = c ⊤ ˆxDW . Since ˆxDW must lie within P ′ ⊆ Q ′ and also within Q ′′ , this shows that zDW ≥ zLP . 3.3 Cutting Plane Method Although the cutting plane method =-=[16, 7, 30, 20]-=- is not usually thought of as a decomposition method, it can in fact be viewed as such. Under our assumption that the separation problem associated with P ′ can be solved effectively, a cutting plane ... |

32 |
Lagrangean relaxation
- Beasley
- 1993
(Show Context)
Citation Context ... principle of decomposition. In Section 4, we show that these methods are very closely related. 3.1 Lagrangian Relaxation For a given vector of dual multipliers u ∈ Rm′′ + , the Lagrangian relaxation =-=[9, 6, 28, 23]-=- of (IP) is given by time. zLR(u) = min s∈F ′{(c⊤ − u ⊤ A ′′ )s + u ⊤ b ′′ }. (LR) 1 The term practical is not defined rigorously, but denotes an algorithm with a “reasonable” average-case running 2I... |

21 |
A set-partitioning-based exact algorithm for the vehicle routing problem.
- Agarwal, Mathur, et al.
- 1989
(Show Context)
Citation Context |

21 |
Steiner problem in graphs: Lagrangean relaxation and cutting planes
- Lucena
- 1993
(Show Context)
Citation Context ...ting Stock Problem [15, 37], the Traveling Salesman Problem [19], the Generalized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem =-=[25]-=-, the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicle Routing Problem (CVRP) [1, 10, 32, 12]. To expose the desired substructure, a common approach... |

21 |
An integer programming approach to scheduling, in: Computer Scheduling of Public Transport. Urban Passenger Vehicle and Crew Scheduling.
- Ryan, Foster
- 1981
(Show Context)
Citation Context ...at compact model is not always first formulated explicitly [42]. Branch-and-bound approaches in which column generation is used to obtain a bound in each node are known as branch-and-price algorithms =-=[34, 35, 40]-=-. It is easy to verify that (DWLP) is an LP dual of (LDLP), which immediately shows that zDW = zLD (see [31] for a detailed treatment of this fact). Hence, zDW is a valid lower bound on zIP that we s∈... |

18 |
Lot-sizing with start-up times
- Vanderbeck
- 1998
(Show Context)
Citation Context ...dure we call price and cut. When employed as the bounding procedure in a branch-and-bound framework, the overall technique is called branch, price, and cut and has been studied by a number of authors =-=[38, 22, 4, 36]-=-. Simultaneous generation of columns and valid inequalities is difficult in general because the addition of valid inequalities may destroy the structure of the column generation subproblem (for a disc... |

18 | On compact formulations for integer programs solved by column generation. Annals of operations research,
- Villeneuve, Desrosiers, et al.
- 2005
(Show Context)
Citation Context ...natural” formulation, can be seen as arising from the application of Dantzig-Wolfe decomposition to an underlying “compact” model, even if that compact model is not always first formulated explicitly =-=[42]-=-. Branch-and-bound approaches in which column generation is used to obtain a bound in each node are known as branch-and-price algorithms [34, 35, 40]. It is easy to verify that (DWLP) is an LP dual of... |

17 |
Nonsmooth dual methods in integer programming.
- Neame
- 2000
(Show Context)
Citation Context ... principle of decomposition. In Section 4, we show that these methods are very closely related. 3.1 Lagrangian Relaxation For a given vector of dual multipliers u ∈ Rm′′ + , the Lagrangian relaxation =-=[9, 6, 28, 23]-=- of (IP) is given by time. zLR(u) = min s∈F ′{(c⊤ − u ⊤ A ′′ )s + u ⊤ b ′′ }. (LR) 1 The term practical is not defined rigorously, but denotes an algorithm with a “reasonable” average-case running 2I... |

14 |
Facets of the three-index assignment polytope
- Balas, Saltzman
- 1989
(Show Context)
Citation Context ...ity Flow Problem [11], the Cutting Stock Problem [15, 37], the Traveling Salesman Problem [19], the Generalized Assignment Problem [18, 35], the Bin Packing Problem [39], the Axial Assignment Problem =-=[2]-=-, The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicle Routing Problem (CVRP) [1, 10, 32, 12]. To expose the desired ... |

8 | Decomposition in integer programming - Ralphs, Galati - 2005 |

8 |
Efficient Cuts in Lagrangean ’Relax-and-Cut’
- Guignard
- 1998
(Show Context)
Citation Context .... As with both the previously discussed methods, the difficulty with relax and cut is that the valid inequalities generated by separating ˆs from P may not be improving, as Guignard first observed in =-=[17]-=-. As we have already noted, we cannot verify that generated inequalities are violated by an optimal fractional solution, which is the usually employed necessary condition for an inequality to be impro... |

8 | Decomposition and dynamic cut generation in integer linear programming
- Ralphs, Galati
- 2006
(Show Context)
Citation Context ...onds to a basic variable in an optimal solution to (DWLP)). The following theorem states that the set S must contain all members of the decomposition. The proof is straightforward and can be found in =-=[33]-=-. Theorem 2 {s ∈ F ′ | ˆ λs > 0} ⊆ S. In fact, we can show something stronger. The set S is comprised exactly of those members of F ′ corresponding to columns of the (DWLP) with reduced cost zero, so ... |

7 |
Non delayed relax-and-cut algorithms
- Lucena
- 2005
(Show Context)
Citation Context ...ating an arbitrary real vector is difficult, but the problem of separating a solution to a given combinatorial relaxation is easy. This notion has been discussed in the literature in several contexts =-=[32, 26, 3]-=- and leads to a separation technique that can be embedded within price and cut[33]. The idea is to replace direct separation of the fractional point (which may be difficult) with separation of members... |

7 |
A branch and cut algorithm for the solution of large scale traveling salesman problems
- Padberg, Rinaldi
- 1991
(Show Context)
Citation Context |

6 |
An improved dual-based algorithm for the generalized assignment problem
- Guignard, Rosenwein
- 1989
(Show Context)
Citation Context ...roaches have been proposed in the literature includes the Multicommodity Flow Problem [11], the Cutting Stock Problem [15, 37], the Traveling Salesman Problem [19], the Generalized Assignment Problem =-=[18, 35]-=-, the Bin Packing Problem [39], the Axial Assignment Problem [2], The Steiner Tree Problem [25], the Single-Machine Scheduling Problem [36], the Graph Coloring Problem [27], and the Capacitated Vehicl... |