The long-step method of analytic centers for fractional problems

We develop a Tong-step surface-following version of the method of analytic centers for the fractional-linear problem min{to [ toB(x) A ( x ) E H, B (x ) E K, x C G}, where H is a closed convex domain, K is a convex cone contained in the recessive cone of H, G is a convex domain and B (.), A (.) are affine mappings. Tracing a two-dimensional surface of analytic centers rather than the usual path of centers allows to skip the initial "centering" phase of the pathfollowing scheme. The proposed long-step policy of tracing the surface fits the best known overall polynomial-time complexity bounds for the method and, at the same time, seems to be more attractive computationally than the short-step policy, which was previously the only one giving good complexity bounds. © 1997 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.