A necessary and a sufficient condition for the existence of optimizers of linear programming problems

Abstract

Umoren (2000) has given the Linear Exchange Algorithm (LEA) for solving certain class of LP problems using the principles of optimal experimental design. This work sets out to establish, first, a first – order necessary condition for the existence of optimizers (minimizers) of certain class of LP problems. That is, if the d-function at the end point of the kth iteration is less than the d-function at any other point within the experimental space, then the d-function at the minimizer x* is the maximum of the minimum d-functions for k different iterations. Second, we also establish the fact that the differential d-functions between the starting point xok and the end point xk of the kth iteration in the LEA is a non-increasing function. This is a sufficient condition for the existence optimizers of LP problems.



Key Word: Linear Exchange Algorithm, Experimental Design, Linear Programming
Problems, d-function.

[Global Jnl Mathematical Sci Vol.2(1) 2003: 9-16]