Book ChapterDOI
Lagrangian Relaxation for Integer Programming
Arthur M. Geoffrion
- pp 243-281
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It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8].Abstract:
It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8]. They also had a good deal to do with shaping my early career and, consequently, much of what followed.read more
Citations
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Journal ArticleDOI
Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs
Hanif D. Sherali,Gyunghyun Choi +1 more
TL;DR: A class of procedures to recover primal solutions directly from the information generated in the process of using pure or deflected subgradient optimization methods to solve such Lagrangian dual formulations is presented.
Journal ArticleDOI
Convex nondifferentiable optimization: A survey focused on the analytic center cutting plane method
TL;DR: A self-contained convergence analysis that uses the formalism of the theory of self-concordant functions, but for the main results, direct proofs based on the properties of the logarithmic function are given.
Journal ArticleDOI
An exact algorithm for the capacitated facility location problems with single sourcing
TL;DR: A primal heuristic, based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied, is incorporated into the Lagrangian heuristic.
Journal ArticleDOI
The quadratic knapsack problem-a survey
TL;DR: A survey of upper bounds presented in the literature is given, and the relative tightness of several of the bounds shown are compared with respect to strength and computational effort.
Journal ArticleDOI
A survey of algorithms for the single machine total weighted tardiness scheduling problem
TL;DR: Four branch and bound algorithms use lower bounds obtained from the Lagrangean relaxation of machine capacity constraints and from dynamic programming state-space relaxation to minimize total weighted tardiness of jobs on a single machine.
References
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Journal ArticleDOI
The Traveling-Salesman Problem and Minimum Spanning Trees
Michael Held,Richard M. Karp +1 more
TL;DR: It is shown that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers.
Journal ArticleDOI
Validation of subgradient optimization
TL;DR: It is concluded that the “relaxation” procedure for approximately solving a large linear programming problem related to the traveling-salesman problem shows promise for large-scale linear programming.
Journal ArticleDOI
Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources
TL;DR: The use of Lagrange multipliers for optimization in the presence of constraints is not limited to differentiable functions but can be applied to problems of maximizing an arbitrary real valued objective function over any set whatever, subject to bounds on the values of any other finite collection of real valued functions denned on the same set as mentioned in this paper.
Book ChapterDOI
Multicommodity Distribution System Design by Benders Decomposition
Arthur M. Geoffrion,G. W. Graves +1 more
TL;DR: In this paper, a multicommodity capacitated single-period version of the problem is formulated as a mixed integer linear program, and a solution technique based on Benders Decomposition is developed, implemented, and successfully applied to a real problem for a major food firm with 17 commodity classes, 14 plants, 45 possible distribution center sites, and 121 customer zones.
Journal ArticleDOI
The traveling-salesman problem and minimum spanning trees: Part II
Michael Held,Richard M. Karp +1 more
TL;DR: An efficient iterative method for approximating this bound closely from below is presented, and a branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it.