Primal-dual algorithms for linear programming based on the logarithmic barrier method

In this paper we will deal with primal{dual interior point methods for solving the linear programming problem. We present a short step and a long step path{following primal{dual method and derive polynomial{ time bounds for both methods. The iteration bounds are as usual in the existing literature, namely O(pnL) iterations for the short step, and O(nL) for the long step variant. In the analysis of both variants we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long step method strongly depends on the fact that the (usual) search directions form a descent direction for the so{called primal{dual logarithmic barrier function.