Numerical Inclusion of Optimum Point for Linear Programming

概要

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This paper concerns with the following linear programming problem: \[ \mbox{Maximize } c^tx, \mbox{ subject to } Ax \leqq b \mbox{ and } x\geqq 0, \] where $A \in \F^{m\times n}$, $b \in \F^m$ and $c, x \in \F^n$. Here, $\F$ is a set of floating point numbers. The aim of this paper is to propose a numerical method of including an optimum point of this linear programming problem provided that a good approximation of an optimum point is given. The proposed method is base on Kantorovichs theorem and the continuous Newton method. Kantorovichs theorem is used for proving the existence of a solution for complimentarity equation and the continuous Newton method is used to prove feasibility of that solution. Numerical examples show that a computational cost to include optimum point is about 4 times than that for getting an approximate optimum solution.