We develop a long-step surface-following version of the method of analytic centers for the fractional-linear problem min{t0 | t0B(x) – A(x) ∈ H, B(x) ∈ K, x ∈ 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 us to skip the initial ‘centering’ phase of the path-following 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.
