The 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{t ₀ |t ₀ B(x) −A(x) εH, B(x) εK, x εG}, whereH is a closed convex domain,K is a convex cone contained in the recessive cone ofH, G is a convex domain andB(·),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 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. The research was partly supported by the Israeli-American Binational Science Foundation (BSF).