JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: VoL 63, No. 3, DECEMBER 1989 New Trajectory-Following Polynomial-Time Algorithm for Linear Programming Problems 1 C. ROOS 2 Communicated by F. Zirilli Abstract. A new interior point method for the solution of the linear programming problem is presented. It is shown that the method admits a polynomial time bound. The method is based on the use of the trajectory of the problem, which makes it conceptually very simple. It has the advantage above related methods that it requires no problem transformation (either affine or projective) and that the feasible region may be unbounded. More importantly, the method generates at each stage solutions of both the primal and the dual problem. This implies that, contrary to the simplex method, the quality of the present solution is known at each stage. The paper also contains a practical (i.e., deep- step) version of the algorithm. Key Words. Linear programming, interior point methods, path- following methods, polynomial-time algorithms. 1. Introduction In this paper, we present a new interior-point algorithm for the solution of the linear programming problem, based on the use of the trajectory of the problem, and which requires only polynomial time. Since the work of Karmarkar (Ref. 1), a lot of research activity has taken place, just to obtain a better understanding of his interior-point method for linear programming. This has resulted in a large number of research papers (cf. Refs. 2-30). The present paper is a completely rewritten and simplified version of Ref. 21. The list of references contains only a selection and is far from complete. In some of these papers, it is indicated that Karmarkar's method is closely The author is indebted to J. Bisschop, P. C. J. Ivl. Geven, J. H. Van Lint, J. Ponstein, and J. P. Vial for their remarks on an earlier version of this paper. 2 Assistant Professor, Department of Mathematics and Informatics/Computer Science, Delft University of Technology, Delft, Holland. 433 0022.3239/89/~200-043350&00/00 © 1989 P~enum Publishing Corporation