A New Method to Find All Alternative Extreme Optimal Points for Linear Programming Problem

The problem of linear programming (LP) is one of the earliest formulated problems in mathematical programming where a linear function has to be maximized (minimized) over a convex constraint polyhedron X. The simplex algorithm was early suggested for solving this problem by moving toward a solution on the exterior of the constraint polyhedron X. In 1984, the area of linear programming underwent a considerable change of orientation when Karmarker (1984) introduced an algorithm for solving (LP) problems which moves through the interior of the polyhedron. This algorithm of Karmarker's and subsequent additional variants (Adler et al., 1989; Karmarkar, 1984) established a new class of algorithms for solving linear programming problems known as the interior point methods . In the case of some linear programs sometimes the solution is not unique and decisions may be taken based on thesealternatives. In this paper we present a feasible direction method to find all alternative optimal extreme points for the linear programming problem. This method is based on the conjugate gradient projection method for solving non-linear programming problem with linear constraints (Goldfarb, 1969; Goldfarb and Lapiduo, 1968). In section 2 we give a full description of the problem together with our main results while section 3 contains the steps of our new algorithm. An example to illustrate our algorithm is given in section 4 followed by our conclusion in section 5.