Combinatorics Advances pp 207-216 | Cite as

# A Simple Polynomial Time Algorithm for a Convex Hull Problem Equivalent to Linear Programming

## Abstract

Over the rationals, the general linear programming problem is equivalent to the convex hull problem of determining if a given *m* × *n* matrix *H* has a nontrivial nonnegative zero. We give a polynomial time algorithm that either finds a nontrivial nonnegative zero of *H*, or it obtains a hyperplane separating the column vectors of *H* from the origin. In particular, the algorithm provides an alternate proof of a strengthened version of Gordan’s duality theorem, previously proved by the author. The algorithm which is motivated by this duality theorem is analogous to Karmarkar’s algorithm but its analysis is much simpler.

## Keywords

Arithmetic Operation Iteration Algorithm Basic Feasible Solution Recessive Cone Strengthened Version## Preview

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