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Journal ArticleDOI

Decomposition Principle for Linear Programs

George B. Dantzig, +1 more
- 01 Feb 1960 - 
- Vol. 8, Iss: 1, pp 101-111
TLDR
A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations.
Abstract
A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations. The coordinating program generates at each cycle new objective forms for each part, and each part generates in turn from its optimal basic feasible solutions new activities columns for the interconnecting program. Viewed as an instance of a “generalized programming problem” whose columns are drawn freely from given convex sets, such a problem can be studied by an appropriate generalization of the duality theorem for linear programming, which permits a sharp distinction to be made between those constraints that pertain only to a part of the problem and those that connect its parts. This leads to a generalization of the Simplex Algorithm, for which the decomposition procedure becomes a special case. Besides holding promise for the efficient computation of large-scale systems, the principle yields a certain rationale for the “decentralized decision process” in the theory of the firm. Formally the prices generated by the coordinating program cause the manager of each part to look for a “pure” sub-program analogue of pure strategy in game theory, which he proposes to the coordinator as best he can do. The coordinator finds the optimum “mix” of pure sub-programs using new proposals and earlier ones consistent with over-all demands and supply, and thereby generates new prices that again generates new proposals by each of the parts, etc. The iterative process is finite.

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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.
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Decomposition methods in stochastic programming

TL;DR: Basic ideas of cutting plane methods, augmented Lagrangian and splitting methods, and stochastic decomposition methods for convex polyhedral multi-stage stochastically programming problems are reviewed.
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Optimal Programming of Lot Sizes, Inventory and Labor Allocations

TL;DR: In this article, the application of the Dantzig and Wolfe decomposition principle and a method for creating alternative set-up sequences as they are needed by means of a computation of the Wagner and Whitin type is described as a method to overcome the computational difficulty.