## A technique for speeding up the solution of the Lagrangean dual (1991)

### Cached

### Download Links

- [web.mit.edu]
- [web.mit.edu]
- [web.mit.edu]
- [web.mit.edu]
- [dspace.mit.edu]
- [dspace.mit.edu]
- DBLP

### Other Repositories/Bibliography

Citations: | 10 - 0 self |

### BibTeX

@MISC{Bertsimas91atechnique,

author = {Dimitris Bertsimas and James B. Orlin},

title = {A technique for speeding up the solution of the Lagrangean dual},

year = {1991}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

1159 |
Integer and Combinatorial Optimization
- Nemhauser, Wolsey
- 1988
(Show Context)
Citation Context ... then the solution (yo, o) is feasible in (D1) and thus we can start Vaidya's sliding objective algorithm (see [29]) or the Sliding objective Ellipsoid algorithm (see for example Nemhauser and Wolsey =-=[25]-=-, p.155) with the same separation problem. If u < o0, a new extreme point is generated, which is added to (Di). If, on the other hand, the subproblem is unbounded, then a new extreme ray of S is gener... |

847 |
Matrix multiplication via arithmetic progressions
- Coppersmith, Winograd
- 1990
(Show Context)
Citation Context ...rations to solve the separation problem (SEP), L is th input size of problem (P1 ) and M(k) is the number of arithmetic operations for multipl*ing two h x k matrices. (It is known that M(k) = O(k2-3) =-=[6]-=-). For comparison, the number of iterations of the ellipsoid algorithm is O(k 2 L) and its running time is O(Tk 2 L + k 4 L). Note that the Ellipsoid algorithm does not benefit from fast matrix multip... |

263 |
The Lagrangian relaxation method for solving integer programming problems
- FISHER
- 1981
(Show Context)
Citation Context ...ing and Lagrangean relaxations can play an important role in solving many problems met in practice. The landmark in the development of Lagrangian relaxation (see for example, Geoffrion [12] or Fisher =-=[11]-=-) for combinatorial optimization problems were the two papers for the traveling salesman problem (TSP) by Held and Karp [16], [17]. In the first paper, Held and Karp [16] proposed a Lagrangian relaxat... |

240 | Fast approximation algorithms for fractional packing and covering problems - Plotkin, Shmoys, et al. - 1995 |

130 |
The traveling-salesman problem and minimum spanning trees
- Held, Karp
- 1971
(Show Context)
Citation Context ...t of Lagrangian relaxation (see for example, Geoffrion [12] or Fisher [11]) for combinatorial optimization problems were the two papers for the traveling salesman problem (TSP) by Held and Karp [16], =-=[17]-=-. In the first paper, Held and Karp [16] proposed a Lagrangian relaxation based on the notion of 1-tree for the TSP. Using a complete characterization of the 1-tree polytope, they showed that this Lag... |

122 |
A New Algorithm for Minimizing Convex Functions Over Convex Sets
- Vaidya
- 1996
(Show Context)
Citation Context ...tion value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm [20] or the recent algorithm of Vaidya =-=[29]-=-. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Suprisingly, our method is significantly faster than interior point me... |

118 |
Validation of Subgradient Optimization
- Held, Wolfe, et al.
- 1974
(Show Context)
Citation Context ... a classical formulation of the TSP. In the second paper, Held and Karp [17] introduced a method, which is now known under the name of subgradient optimization 2 IIIs(see also Held, Wolfe and Crowder =-=[18]-=-), to solve the Lagrangian dual. Despite its success in computational experiments subgradient optimization is not known to be a polynomial method. In this paper we propose new techniques for the solut... |

118 |
A polynomial algorithm for linear programming
- Khachian
- 1979
(Show Context)
Citation Context ...s. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm =-=[20]-=- or the recent algorithm of Vaidya [29]. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Suprisingly, our method is sign... |

111 |
R.T.: Network design and transportation planning: models and algorithms
- Magnanti, Wong
- 1984
(Show Context)
Citation Context ...rtation, distribution, communication, and several other problem settings that make basic cost tradeoffs between operating costs and fixed costs for providing network facilities (see Magnanti and Wong =-=[24]-=-). The problem is described as follows. We are given a set of nodes N, a set of uncapacitated arcs A and a set K of commodities. For each k E K, one unit of flow of commodity k must be sent from its o... |

89 | A Cutting Plane Algorithm for the Linear Ordering Problem - Grötschel, Jünger, et al. - 1984 |

70 |
An application of combinatorial optimization to statistical physics and circuit layout design
- Barahona, Grotschel, et al.
- 1988
(Show Context)
Citation Context ...ing problems (Johnson, Kostreva and Suhl [19]), production planning problems with changeover costs (Magnanti and Vachani [23]), and certain spin glass problems that arise in physics (Barahona et. al. =-=[4]-=-). These successes show that linear programming and Lagrangean relaxations can play an important role in solving many problems met in practice. The landmark in the development of Lagrangian relaxation... |

64 |
Optimization of a 532-City Symmetric Traveling Salesman Problem by Branch-and-Cut
- Padberg, Rinaldi
- 1987
(Show Context)
Citation Context ... Padberg [8] and Padberg and Hong [26], researchers have now been able to solve to optimality a number of applications including traveling salesman problems with up to 2000 nodes (Padberg and Rinaldi =-=[27]-=-), a variety of large scale (up to about 2000 integer variables) real world zero-one business planning problems (Crowder, Johnson and Padberg [7]), input-output matrices that arise in economic plannin... |

46 | Survivable networks, linear programming relaxations and the parsimonious property
- Bertsimas, Goemans
- 1993
(Show Context)
Citation Context ... then our approach leads to O(n 4 L) time complexity while Vaidya's or the Ellipsoid method take O(n 8 L), a savings of O(n 4 ). 3.2 The Steiner tree and the k-connected problem Goemans and Bertsimas =-=[13]-=- prove that under the triangle inequality, the cost of the LP relaxation of the Steiner tree problem z,uie, and the k-connected problem Zk-,on are related in the following way 1 k ZStener = ZHK, zk-co... |

42 | Improved algorithms for bipartite network flow
- Ahuja, Orlin, et al.
- 1994
(Show Context)
Citation Context ...r, tree, unbalanced bipartite, etc.) for which the related network flow problem can be solved faster than in a general graph. For example, for network flow problems on unbalanced bipartite graphs see =-=[2]-=-. Assume that BI = K. The network flow problem can be formulated as follows: subject to z = Min cAA + CBZB (39) NAZA + NBXB = b (40) Q< A < A (41) 0 z < (42) where NA, NB is the arc incidence matrix c... |

39 |
Lagrangean relaxation and its uses in integer programming
- Geoffrion
- 1974
(Show Context)
Citation Context ...linear programming and Lagrangean relaxations can play an important role in solving many problems met in practice. The landmark in the development of Lagrangian relaxation (see for example, Geoffrion =-=[12]-=- or Fisher [11]) for combinatorial optimization problems were the two papers for the traveling salesman problem (TSP) by Held and Karp [16], [17]. In the first paper, Held and Karp [16] proposed a Lag... |

35 |
Solving large scale zero-one linear programming problems
- Crowder, Johnson, et al.
- 1983
(Show Context)
Citation Context ...n problems with up to 2000 nodes (Padberg and Rinaldi [27]), a variety of large scale (up to about 2000 integer variables) real world zero-one business planning problems (Crowder, Johnson and Padberg =-=[7]-=-), input-output matrices that arise in economic planning (Gr6tschel, Jinger and Reinelt [14]), multiple item capacitated lot size problems that arise in production planning (Barany, Van Roy and Wolsey... |

35 |
A dual-ascent procedure for large-scale uncapacitated network design
- Balakrishnan, Magnanti, et al.
- 1989
(Show Context)
Citation Context ... tree, uncapacitated plant location, traveling salesman and Steiner tree problems. There are a number of IP formulations for the problem. For a review, see Magnanti and Wong [24]. Balakrishnan et al. =-=[3]-=- propose the following multicommodity flow formulation, which contains two types of variables, one modeling discrete design choices and the other continuous flow decisions. Let yij be a binary variabl... |

31 |
Testing membership in matroid polyhedra
- Cunningham
- 1984
(Show Context)
Citation Context ...m. Overall this approach leads to O(K(nL) 2 [T(n) + M(n)]) arithmetic operations. Alternatively in order to solve the separation problem, one could use the strongly polynomial algorithm of Cunningham =-=[9]-=-, but unfortunately the running time would not be as good. Overall, we expect that our approach will work better, in problems in which the separation problem is much harder than the optimization probl... |

30 |
On the symmetric travelling salesman problem: a computational study
- Padberg, Hong
- 1980
(Show Context)
Citation Context ...rogramming methodology (LP relaxations, polyhedral combinatorics) have been very impressive. Starting with the seminal computational research conducted by Crowder and Padberg [8] and Padberg and Hong =-=[26]-=-, researchers have now been able to solve to optimality a number of applications including traveling salesman problems with up to 2000 nodes (Padberg and Rinaldi [27]), a variety of large scale (up to... |

25 | Finding minimum-cost flows by double scaling
- Ahuja, Goldberg, et al.
- 1992
(Show Context)
Citation Context ...n problem is a min-cost flow problem. Under the similarity assumption, the time to solve the separation problem is T = O(nmlog 2 n). For a more refined definition of T see, for example, Ahuja et. al. =-=[1]-=-. As a result, applying theorem 2 we can solve the multicommodity flow problem in o([Kmnlog 2 n + M(m + K)](m + K)L) = O([Knm 2 log 2 n + m 3 38]L). The comparison for multicomodity flows is more comp... |

24 |
Speeding-up linear programming using fast matrix multiplication
- Vaidya
- 1989
(Show Context)
Citation Context ...lve the multicommodity flow problem in o([Kmnlog 2 n + M(m + K)](m + K)L) = O([Knm 2 log 2 n + m 3 38]L). The comparison for multicomodity flows is more complex because of another algorithm of Vaidya =-=[28]-=-. He considered the problem in which each commodity consists of a single source and a single sink. The resulting running time is O(K 2 5 n 2 / L). This running time dominates ours in some cases and is... |

24 |
Uncapacitated lot sizing: the convex hull of solutions. Math Program Stud 22: 32–43
- Barany, TJ, et al.
- 1984
(Show Context)
Citation Context ... input-output matrices that arise in economic planning (Gr6tschel, Jiinger and Reinelt [14]), multiple item capacitated lot size problems that arise in production planning (Barany, Van Roy and Wolsey =-=[5]-=- , Eppen and Martin [10], Leung, Magnanti and Vachani [21]), strategic planning problems (Johnson, Kostreva and Suhl [19]), production planning problems with changeover costs (Magnanti and Vachani [23... |

14 |
Tailoring benders decomposition for uncapacited network design. Mathematical programming study
- Magnanti, Mireault, et al.
- 1986
(Show Context)
Citation Context ...Although the network design problem is a hard discrete optimization problem, in the last decade researchers have proposed several computationally successful approaches for solving it. Magnanti et al. =-=[22]-=- propose a Benders decomposition approach, and Balakrishnan et al. [3] propose a dual ascent heuristic which has solved large instances of network design problems to within 2%-5% of optimality. In bot... |

12 |
Facets and algorithms for capacitated lot sizing
- Leung, Magnanti, et al.
- 1989
(Show Context)
Citation Context ...Gr6tschel, Jinger and Reinelt [14]), multiple item capacitated lot size problems that arise in production planning (Barany, Van Roy and Wolsey [5] , Eppen and Martin [10], Leung, Magnanti and Vachani =-=[21]-=-), strategic planning problems (Johnson, Kostreva and Suhl [19]), production planning problems with changeover costs (Magnanti and Vachani [23]), and certain spin glass problems that arise in physics ... |

11 |
A strong cutting plane algorithm for production scheduling with changeover costs," working paper OR173-87
- Magnanti, Vachani
- 1989
(Show Context)
Citation Context ...[5] , Eppen and Martin [10], Leung, Magnanti and Vachani [21]), strategic planning problems (Johnson, Kostreva and Suhl [19]), production planning problems with changeover costs (Magnanti and Vachani =-=[23]-=-), and certain spin glass problems that arise in physics (Barahona et. al. [4]). These successes show that linear programming and Lagrangean relaxations can play an important role in solving many prob... |

10 |
Solving large scale symmetric traveling salesman problems to opumality," Management Seienee 26
- Crowder, Padberg
- 1980
(Show Context)
Citation Context ...ccesses in using linear programming methodology (LP relaxations, polyhedral combinatorics) have been very impressive. Starting with the seminal computational research conducted by Crowder and Padberg =-=[8]-=- and Padberg and Hong [26], researchers have now been able to solve to optimality a number of applications including traveling salesman problems with up to 2000 nodes (Padberg and Rinaldi [27]), a var... |

8 |
Suhl: Solving 0-1 integer programming problems arising from large scale planning models
- Johnson, Kostreva, et al.
(Show Context)
Citation Context ...lot size problems that arise in production planning (Barany, Van Roy and Wolsey [5] , Eppen and Martin [10], Leung, Magnanti and Vachani [21]), strategic planning problems (Johnson, Kostreva and Suhl =-=[19]-=-), production planning problems with changeover costs (Magnanti and Vachani [23]), and certain spin glass problems that arise in physics (Barahona et. al. [4]). These successes show that linear progra... |

6 |
Solving multi-item capacitated lot sizing problems using variable redefinition
- Eppen, Martin
- 1987
(Show Context)
Citation Context ... that arise in economic planning (Gr6tschel, Jinger and Reinelt [14]), multiple item capacitated lot size problems that arise in production planning (Barany, Van Roy and Wolsey [5] , Eppen and Martin =-=[10]-=-, Leung, Magnanti and Vachani [21]), strategic planning problems (Johnson, Kostreva and Suhl [19]), production planning problems with changeover costs (Magnanti and Vachani [23]), and certain spin gla... |

5 | Geometric Algorithms and Combinatorial Optimization - Gr6tschel, Lovasz, et al. - 1988 |

4 |
A cutting plane algorithm for the linear ordering problem
- GrStschel, Jiinger, et al.
- 1984
(Show Context)
Citation Context ... about 2000 integer variables) real world zero-one business planning problems (Crowder, Johnson and Padberg [7]), input-output matrices that arise in economic planning (Gr6tschel, Jiinger and Reinelt =-=[14]-=-), multiple item capacitated lot size problems that arise in production planning (Barany, Van Roy and Wolsey [5] , Eppen and Martin [10], Leung, Magnanti and Vachani [21]), strategic planning problems... |

4 | A new Lagrangean relaxation approach to the generalized assignment problem - Jornsten, Nasberg - 1986 |

1 |
An application of combinatorial optimization to statistical physics and circuit layout design
- Bahona, Gr6tschel, et al.
- 1988
(Show Context)
Citation Context ...ing problems (Johnson, Kostreva and Suhl [19]), production planning problems with changeover costs (Magnanti and Vachani [23]), and certain spin glass problems that arise in physics (Barahona et. al. =-=[4]-=-). These successes show that linear programming and Lagrangean relaxations can play an important role in solving many problems met in practice. The landmark in the development of Lagrangian relaxation... |

1 |
Uncapacitated lot sising: the convex hull of solutions
- Barany, Roy, et al.
- 1984
(Show Context)
Citation Context ..., input-output matrices that arise in economic planning (Gr6tschel, Jinger and Reinelt [14]), multiple item capacitated lot size problems that arise in production planning (Barany, Van Roy and Wolsey =-=[5]-=- , Eppen and Martin [10], Leung, Magnanti and Vachani [21]), strategic planning problems (Johnson, Kostreva and Suhl [19]), production planning problems with changeover costs (Magnanti and Vachani [23... |

1 |
A cutting plane algorithm for the linear si prmlem
- Oi&W6tsche, Reinelt
- 1984
(Show Context)
Citation Context ...o about 2000 integer variables) real world zero-one business planning problems (Crowder, Johnson and Padberg [7]), input-output matrices that arise in economic planning (Gr6tschel, Jinger and Reinelt =-=[14]-=-), multiple item capacitated lot size problems that arise in production planning (Barany, Van Roy and Wolsey [5] , Eppen and Martin [10], Leung, Magnanti and Vachani [21]), strategic planning problems... |

1 |
Solving 0-1 integer programming problerns arising from large scale planning models
- Johnson, Kostreva, et al.
- 1985
(Show Context)
Citation Context ...lot size problems that arise in production planning (Barany, Van Roy and Wolsey [5] , Eppen and Martin [10], Leung, Magnanti and Vachani [21]), strategic planning problems (Johnson, Kostreva and Suhl =-=[19]-=-), production planning problems with changeover costs (Magnanti and Vachani [23]), and certain spin glass problems that arise in physics (Barahona et. al. [4]). These successes show that linear progra... |

1 | Finding minimum-cost flows by double scaling," to appear in - Ahuja, Goldberg, et al. - 1991 |

1 | On the complexity of Lagrange multiplier search," unpublished manuscript - Chandru, Trick |

1 | S°lvinglarge scale zer°-°nelinear pr°grammingpr°blems - Crowder, Johnson, et al. - 1983 |

1 | A new algorithm for minimizing convex fnnctions over convex sets - Vaidya - 1990 |