# Lagrangian Relaxation

@inproceedings{Lemarchal2001LagrangianR, title={Lagrangian Relaxation}, author={C. Lemar{\'e}chal}, booktitle={Computational Combinatorial Optimization}, year={2001} }

Lagrangian relaxation is a tool to find upper bounds on a given (arbitrary) maximization problem. Sometimes, the bound is exact and an optimal solution is found. Our aim in this paper is to review this technique, the theory behind it, its numerical aspects, its relation with other techniques such as column generation.

#### 330 Citations

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- Computer Science
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Modified Lagrangian bounds are proposed for the generalized assignment problem based on a double decomposable structure of the formulation and a family of greedy heuristics is considered to get feasible solutions. Expand

Refinement of Lagrangian bounds in optimization problems

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Lagrangian constraint relaxation and the corresponding bounds for the optimal value of an original optimization problem are examined. Techniques for the refinement of the classical Lagrangian bounds… Expand

A Lagrangian bound for many-to-many assignment problems

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- J. Comb. Optim.
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A simple procedure to tighten the Lagrangian bounds is proposed, and the new bounds are illustrated by a small example and studied numerically for a class of the generalized assignment problems. Expand

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The theory assessing the Lagrangian relaxation approach in the framework of combinatorial optimization indicates that very little can be expected in theory, even though fairly good practical results have been obtained for the unit-commitment problem. Expand

The omnipresence of Lagrange

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- 2007

This paper reviews Lagrangian relaxation from both points of view of theory and algorithms (to dualize a given problem and solve the dual by nonsmooth optimization). Expand

Comparison of Lagrangian bounds for one class of generalized assignment problems

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- 2008

Classical and modified Lagrangian bounds for the optimal value of optimization problems with a double decomposable structure are examined. For the class of generalized assignment problems, this… Expand

PRIMAL ERROR BOUNDS BASED ON THE AUGMENTED LAGRANGIAN AND LAGRANGIAN RELAXATION ALGORITHMS

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- 2006

For a given iterate generated by the augmented Lagrangian or the Lagrangian relaxation based method, we derive estimates for the distance to the primal solution of the underlying optimization… Expand

Calculating the best dual bound for problems with multiple Lagrangian relaxations

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An optimization problem over the set of Lagrangian relaxations with the objective to indicate the relaxation producing the best dual bound and an iterative technique to solve this problem is proposed based on constraints generation scheme. Expand

Studying properties of Lagrangian bounds for many-to-many assignment problems

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Classical and modified Lagrangian bounds for the optimal value of optimization problems with a double decomposable structure are studied. For the class of many-to-many assignment problems, this… Expand

Lagrangian heuristic for a class of the generalized assignment problems

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A greedy heuristic is applied to get a feasible Lagrangian-based solution for many-to-many assignment problems taking into account capacity limits for task and agents and a numerical study is presented to demonstrate the efficiency. Expand

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