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An algorithmic proof of the polyhedral decomposition theorem

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  • Mustafa Akgül

Abstract

It is well‐known that any point in a convex polyhedron P can be written as the sum of a convex combination of extreme points of P and a non‐negative linear combination of extreme rays of P. Grötschel, Lovász, and Schrijver gave a polynomial algorithm based on the ellipsoidal method to find such a representation for any x in P when P is bounded. Here we show that their algorithm can be modified and implemented in polynomial time using the projection method or a simplex‐type algorithm : in n(2n + 1) simplex pivots, where n is the dimension of x. Extension to the unbounded case is immediate.

Suggested Citation

  • Mustafa Akgül, 1988. "An algorithmic proof of the polyhedral decomposition theorem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(5), pages 463-472, October.
    • Handle: RePEc:wly:navres:v:35:y:1988:i:5:p:463-472
    DOI: 10.1002/1520-6750(198810)35:53.0.CO;2-5
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    1. Peter B. R. Hazell & Carlos Pomareda, 1981. "Evaluating Price Stabilization Schemes with Mathematical Programming," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 63(3), pages 550-556.
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