Interior point method, based on rank-1 updates, for linear programming

We propose a polynomial time primal-dual potential reduction algorithm for linear programming. The algorithm generates sequences dk and vk rather than a primal-dual interior point (xk, sk), where dik = rootxik/sik and vik = rootxiksik for i = 1, 2, ..., n. Only one element of dk is changed in each iteration, so that the work per iteration is bounded by O(mn) using rank-1 updating techniques. The usual primal-dual iterates xk and sk are not needed explicitly in the algorithm, whereas dk and vk are iterated so that the interior primal-dual solutions can always be recovered by aforementioned relations between (xk, sk) and (dk, vk) with improving primal-dual potential function values. Moreover, no approximation of dk is needed in the computation of projection directions.