An Isometric Plane Method for Linear Programming

Abstract

In this paper the following canonical form of a general LP problem, $${\rm max} \ Z=C^TX,$$
$${\rm subject} \ {\rm to} \ AX\geq b$$is considered for $X\in R^n$. The constraints form an arbitrary convex polyhedron $\Omega^m$ in $R^n$. In $\Omega^m$ a strictly interior point is successively moved to a higher isometric plane from a lower one along the gradient function value maximum is found or the infinite value of the objective function is concluded. For an $m\ast n$ matrix $A$ the arithmetic operations of a movement are $O(mn)$ in our algorithm. The algorithm enables one to solve linear equations with ill conditions as well as a general LP problem. Some interesting examples illustrate the efficiency of the algorithm.