## Stabilized Column Generation (1997)

Venue: | DISCRETE MATH |

Citations: | 33 - 3 self |

### BibTeX

@ARTICLE{Merle97stabilizedcolumn,

author = {Olivier du Merle and Daniel Villeneuve and Jacques Desrosiers and Pierre Hansen},

title = {Stabilized Column Generation},

journal = {DISCRETE MATH},

year = {1997},

volume = {194},

pages = {229--237}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

249 |
Partitioning Procedures for Solving Mixed-Variables Programming
- Benders
- 1962
(Show Context)
Citation Context ...roduction Consider a feasible and bounded linear program P and its dual D: min c T x max b T �� (P ) s.t. Ax = b (D) s.t. A T ��sc: xs0 When P has many more variables than D, as when applying =-=Benders [2]-=- or Dantzig-Wolfe [5] decomposition, it is usually solved by column generation. This amounts to applying Kelley's algorithm [15] to D and convergence is often slow. Such undesirable behavior is also o... |

143 |
Optimization theory for large systems
- Lasdon
- 1970
(Show Context)
Citation Context ...T i x i = b i ; x is0; i 2 f1; : : : ; pg ) ; where e i is a unit vector of the same dimension as x i . An estimate of the quality for ~ �� k can be obtained via the computation of a lower bound o=-=n P [17], v(P ; ~ �� k )-=- = b T 0 ~ �� k 0 + p X i=1 b i (c i \Gammasa T i ~ �� k 0 ); where ~ �� k 0 is the vector of components of ~ �� k associated with the first set of constraints P p i=1 A i x i = b 0 . ... |

116 |
The decomposition algorithm for linear programming
- Dantzig, Wolfe
- 1961
(Show Context)
Citation Context ...feasible and bounded linear program P and its dual D: min c T x max b T �� (P ) s.t. Ax = b (D) s.t. A T ��sc: xs0 When P has many more variables than D, as when applying Benders [2] or Dantzi=-=g-Wolfe [5]-=- decomposition, it is usually solved by column generation. This amounts to applying Kelley's algorithm [15] to D and convergence is often slow. Such undesirable behavior is also observed when one trie... |

92 |
The cutting-plane method for solving convex programs
- Kelley
- 1960
(Show Context)
Citation Context ...�sc: xs0 When P has many more variables than D, as when applying Benders [2] or Dantzig-Wolfe [5] decomposition, it is usually solved by column generation. This amounts to applying Kelley's algorith=-=m [15]-=- to D and convergence is often slow. Such undesirable behavior is also observed when one tries to prove optimality of a degenerate solution of P . In other words, many iterations are necessary to obta... |

68 |
Variable neighbourhood search
- Hansen, Mladenović
- 2003
(Show Context)
Citation Context ...gives very rough estimates of the dual variables, the objective function value being around 100 000. On the other hand, the vector parameters ffl \Gamma and ffl + are selected at random in the ranges =-=[9; 11]-=- and [0; 10 \Gamma4 ], respectively, and are kept fixed throughout the solution process. The first interval, for selecting ffl \Gamma , allows for overcovering the flight legs and simulates a Set Cove... |

47 |
Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods, volume 306 of Grundlehren der mathematischen Wissenschaften
- Hiriart-Urruty, Lemarechal
- 1993
(Show Context)
Citation Context ...studies where stabilized column generation leads to large speedup factors or to solution of much larger instances than done before. Stabilization in nonlinear programming has been extensively studied =-=[16, 14]-=-. The originality of the present paper is to propose a linear stabilization scheme which remains entirely within the column generation framework and to demonstrate its efficiency. 2 Problem ~ P Define... |

25 |
A Note on Solving Large p-Median Problems
- Beasley
- 1985
(Show Context)
Citation Context ...erpart of the multisource Weber problem, location of facilities being restricted to a given discrete set of points. The largest problems solved in the literature have ns900 users and ps200 facilities =-=[1]-=-. A column generation algorithm for the p-median, with a very simple oracle, was proposed some time ago [9] but could only solve small instances. Using stabilized column generation coupled with a Vari... |

21 |
Crew Pairing at Air France
- Desaulniers
- 1997
(Show Context)
Citation Context ...formulated as a Set Partitioning Problem. A column generation algorithm is used to find lower bounds in a branch&bound framework, the oracle consisting in a resource constrained shortest path problem =-=[6]-=-. Precise estimates for the optimal values of dual variables are difficult to obtain but a good approximation of the objective function value can be used to compute an average value for them. The clas... |

13 | Improvements and comparison of heuristics for solving the multisource Weber problem
- Brimberg, Hansen, et al.
(Show Context)
Citation Context ...on of problems with ns1060 and ps100 using the following strategies. A heuristic solution, often optimal or very close to the optimum, is first obtained with a Variable Neighbourhood Search heuristic =-=[18, 11, 3]-=-. Let c 1 , c 2 , . . . , c n , denote the index sets of the corresponding user partition and f(c 1 ), f(c 2 ), . . . , f(c n ) their value. Then, given k 2 c i , initial ranges for the dual variables... |

13 |
Constrained nonlinear programming
- Gill, Murray, et al.
- 1989
(Show Context)
Citation Context ...d surplus and slack variables: (P ffl ) min (x;y \Gamma ;y + )0 fc T x : Ax \Gamma y \Gamma + y + = b; y \Gammasffl \Gamma ; y +sffl + g: Alternatively, one could also use exact penalties in l 1 norm =-=[10]-=-, as in nonlinear programming, to narrow the domain of D. Solving P is then equivalent to solving the following problem, with a suitably chosen scalar ffis0: (P ffi ) min x0 c T x + ffi kAx \Gamma bk ... |

13 |
An optimal method for solving the (generalized) multi-Weber problem
- Rosing
- 1992
(Show Context)
Citation Context ...is closest facility. The case p = 2 can easily be solved by d.-c. programming [4] but only small instances are solved exactly for ps3 (i.e., ns50 for p = 3 [4], ns30 for p = 4; 5 and n = 25 for p = 6 =-=[20]-=-). The problem can be expressed as a Set Partitioning Problem, with columns associated with all possible subsets of users and costs corresponding to those of single facility Weber problems with these ... |

10 |
Variable Neighbourhood Search for the p-Median
- Hansen, Mladenović
- 1997
(Show Context)
Citation Context ...the p-median, with a very simple oracle, was proposed some time ago [9] but could only solve small instances. Using stabilized column generation coupled with a Variable Neighbourhood Search heuristic =-=[12]-=- led to solve the continuous relaxation of instances with n = 3038 and ps100. Embedding of this approach in a branch&bound procedure is under way. Results obtained on a SUN SPARC 20 workstation are gi... |

9 |
An algorithm for the m-median plant location problem
- Garfinkel, Neebe, et al.
- 1974
(Show Context)
Citation Context ...gives very rough estimates of the dual variables, the objective function value being around 100 000. On the other hand, the vector parameters ffl \Gamma and ffl + are selected at random in the ranges =-=[9; 11]-=- and [0; 10 \Gamma4 ], respectively, and are kept fixed throughout the solution process. The first interval, for selecting ffl \Gamma , allows for overcovering the flight legs and simulates a Set Cove... |

8 |
Solution of the Multisource Weber and Conditional Weber Problems by D.-C. Programming
- Chen, Hansen, et al.
- 1998
(Show Context)
Citation Context ...ously the locations of p facilities in this plane in order to minimize the sum of (weighted) distances from each user to his closest facility. The case p = 2 can easily be solved by d.-c. programming =-=[4]-=- but only small instances are solved exactly for ps3 (i.e., ns50 for p = 3 [4], ns30 for p = 4; 5 and n = 25 for p = 6 [20]). The problem can be expressed as a Set Partitioning Problem, with columns a... |

5 |
The minisum and minimax location problems revisited
- Hansen, Peeters, et al.
- 1985
(Show Context)
Citation Context ... algorithm can be used to find lower bounds in a branch&bound framework, the oracle consisting in a single facility Weber problem with limited distances [8], solved by a variant of the bsss algorithm =-=[13]-=-. Problems with ns287 and ps100 can then be solved [16]. Stabilizing this algorithm allows solution of problems with ns1060 and ps100 using the following strategies. A heuristic solution, often optima... |

4 |
Crew Pairing for a Regional Carrier
- Desaulniers, Desrosiers, et al.
- 1997
(Show Context)
Citation Context ... Table 1: Instance of an airline crew pairing problem---variations on ffi 0 As an example, we used the stabilized algorithm to solve the linear relaxation of a 986-leg instance for a regional carrier =-=[7]-=-: the number of column generation iterations is reduced by a factor of 3.33 and the cpu time by a factor of 7.41. On the one hand, the strategy for the update-ffi procedure is to update the vector par... |

4 |
G.O.: The facility location problem with limited distances
- Drezner, Mehrez, et al.
- 1991
(Show Context)
Citation Context ...s with these sets of users. A column generation algorithm can be used to find lower bounds in a branch&bound framework, the oracle consisting in a single facility Weber problem with limited distances =-=[8]-=-, solved by a variant of the bsss algorithm [13]. Problems with ns287 and ps100 can then be solved [16]. Stabilizing this algorithm allows solution of problems with ns1060 and ps100 using the followin... |

2 |
Extensions du probl`eme de Weber. Ph.D. dissertation, ' Ecole Polytechnique de
- Krau
- 1997
(Show Context)
Citation Context ...studies where stabilized column generation leads to large speedup factors or to solution of much larger instances than done before. Stabilization in nonlinear programming has been extensively studied =-=[16, 14]-=-. The originality of the present paper is to propose a linear stabilization scheme which remains entirely within the column generation framework and to demonstrate its efficiency. 2 Problem ~ P Define... |