The convergence of a modified barrier method for convex programming

We show, using elementary considerations, that a modified barrier function method for the solution of convex programming problems converges for any fixed positive setting of the barrier parameter. With mild conditions on the primal and dual feasible regions, we show how to use the modified barrier function method to obtain primal and dual optimal solutions, even in the presence of degeneracy. We illustrate the argument for convergence in the case of linear programming, and then generalize it to the convex programming case.