ON SELF-CONCORDANT BARRIER FUNCTIONS FOR CONIC HULLS AND FRACTIONAL-PROGRAMMING

Given a self-concordant barrier function for a convex set J, we determ ine a self-concordant barrier function for the conic hull (J) over til de of J. As our main result, we derive an ''optimal'' barrier for (J) over tilde based on the barrier function for J. Important applications of this result include the conic reformulation of a convex problem, a nd the solution of fractional programs by interior-point methods. The problem of minimizing a convex-concave fraction over some convex set c an be solved by applying an interior-point method directly to the orig inal nonconvex problem, or by applying an interior-point method to an equivalent convex reformulation of the original problem. Our main resu lt allows to analyze the second approach showing that the rate of conv ergence is of the same order in both cases.