Long-Step Method of Analytic Centers for Fractional Problems

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 twodimensional surface of analytic centers rather than the usual path of centers allows 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.