GPM ANALYSIS IN COMBINING INTERIOR POINTS AND THE SIMPLEX METHOD

Abstract

By geometric viewpoint in search directions of linear programming problems (LP), there are two major approaches: the simplex method and the interior-point algorithm originated from Karmarkar`s approach. Many of their variants developed both in theory and applications are still in progress. Roughly speaking, the main difference among them is that the simplex method is devoted for each the exact optimal solution while Karmarkar-based method is computationally fast in approaching to the neighborhood of the optimal solution, but it becomes slow near the optimal point. By the separating hyperplane theorem, we know that the optimal solution of a linear programming problem would locate at the boundary point. However, Karmarkar`s algorithm is claimed as an interior-point approach which takes a solution trajectory path through the interior of the feasible region. On the other hand, these algorithms will coincide the zig-zag situation following the trajectory path before attaining the optimal point meanwhile it cannot achieve the optimal point. Therefore, the purpose of this paper is to study a possible approach by combining the interior point and the simplex method. By means of QR factorization, we give a general analysis on Gradient Projection Method(GPM) in solving linear programming problems. Moreover, we prove that a general assumption-technique constraint matrix A of full rank can be relaxed by using our approach. Finally, we analyze the complexity of this approach to ensure that its upper bound is O(n3).