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

Abstract

We propose a polynomial time primal—dual potential reduction algorithm for linear programming. The algorithm generates sequencesd ^k andv ^k rather than a primal—dual interior point (x ^k,s ^k), where$d_i^k = \sqrt {{{x_i^k } \mathord{\left/ {\vphantom {{x_i^k } {s_i^k }}} \right. \kern-\nulldelimiterspace} {s_i^k }}}$ and$v_i^k = \sqrt {x_i^k s_i^k }$ fori = 1, 2,⋯,n. Only one element ofd ^k 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 iteratesx ^k ands ^k are not needed explicitly in the algorithm, whereasd ^k andv ^k are iterated so that the interior primal—dual solutions can always be recovered by aforementioned relations between (x ^k, s^k) and (d ^k, v^k) with improving primal—dual potential function values. Moreover, no approximation ofd ^k is needed in the computation of projection directions. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.