On self-concordant barrier functions for conic hulls and fractional programming

Abstract

Given a self-concordant barrier function for a convex set[Figure not available: see fulltext.], we determine a self-concordant barrier function for the conic hull[Figure not available: see fulltext.] of[Figure not available: see fulltext.]. As our main result, we derive an "optimal" barrier for[Figure not available: see fulltext.] based on the barrier function for[Figure not available: see fulltext.]. Important applications of this result include the conic reformulation of a convex problem, and the solution of fractional programs by interior-point methods. The problem of minimizing a convex-concave fraction over some convex set can be solved by applying an interior-point method directly to the original nonconvex problem, or by applying an interior-point method to an equivalent convex reformulation of the original problem. Our main result allows to analyze the second approach showing that the rate of convergence is of the same order in both cases.