Journal ArticleDOI
The Lagrangian Relaxation Method for Solving Integer Programming Problems
TLDR
This paper is a review of Lagrangian relaxation based on what has been learned in the last decade and has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering.Abstract:
(This article originally appeared in Management Science, January 1981, Volume 27, Number 1, pp. 1-18, published by The Institute of Management Sciences.)
One of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints. Dualizing the side constraints produces a Lagrangian problem that is easy to solve and whose optimal value is a lower bound (for minimization problems) on the optimal value of the original problem. The Lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm. This approach has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering. This paper is a review of Lagrangian relaxation based on what has been learned in the last decade.read more
Citations
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Journal ArticleDOI
A generalized assignment heuristic for vehicle routing
TL;DR: This paper presents a heuristic for this problem in which an assignment of customers to vehicles is obtained by solving a generalized assignment problem with an objective function that approximates delivery cost and shows that it has outperformed the best existing heuristics on a sample of standard test problems.
Journal ArticleDOI
Rate-distortion methods for image and video compression
TL;DR: An overview of rate-distortion (R-D) based optimization techniques and their practical application to image and video coding is provided and two popular techniques for resource allocation are introduced, namely, Lagrangian optimization and dynamic programming.
Journal ArticleDOI
Emergency Logistics Planning in Natural Disasters
TL;DR: A planning model that is to be integrated into a natural disaster logistics Decision Support System is developed that addresses the dynamic time-dependent transportation problem that needs to be solved repetitively at given time intervals during ongoing aid delivery.
Journal ArticleDOI
Reliability optimization of series-parallel systems using a genetic algorithm
David W. Coit,Alice E. Smith +1 more
TL;DR: A problem-specific genetic algorithm (GA) is developed and demonstrated to analyze series-parallel systems and to determine the optimal design configuration when there are multiple component choices available for each of several k-out-of-n:G subsystems.
References
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Journal ArticleDOI
A Linear Programming Approach to the Cutting-Stock Problem
P. C. Gilmore,Ralph E. Gomory +1 more
TL;DR: In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the cutting-stock problem, which enables one to compute always with a matrix which has no more columns than it has rows.
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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.
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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.
Journal ArticleDOI
A Linear Programming Approach to the Cutting Stock Problem---Part II
P. C. Gilmore,Ralph E. Gomory +1 more
TL;DR: The paper describes a new and faster knapsack method, experiments, and formulation changes, and the introduction of a rational objective function when customers' orders are not for fixed amounts, but rather for a range of amounts.