Decomposition Principle for Linear Programs

TLDR

The decomposition principle treats a linear program as a generalized programming problem with columns drawn from convex sets, enabling a duality-based distinction between part‑specific and inter‑part constraints. The paper proposes a decomposition technique that solves a linear program by iteratively solving linear sub‑programs and a coordinating program derived via linear transformations. The method alternates between a coordinating program that generates new objective forms for each part and sub‑programs that, from their optimal basic feasible solutions, produce new columns for the interconnecting program, generalizing the Simplex algorithm. The principle promises efficient computation of large‑scale systems, provides a rationale for decentralized decision‑making in firms, and yields a finite iterative process where coordinating prices guide parts toward optimal mixes of sub‑programs.

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.