A Polynomial Primal-Dual Path-Following Algorithm for Second-order Cone Programming

Second-order cone programming (SOCP) is the problem of minimizing linear objective function over cross-section of second-order cones and an a ne space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic programming. In this paper we deal with a primal-dual path-following algorithm for SOCP to show many of the ideas developed for primal-dual algorithms for LP and SDP carry over to this problem. We de ne neighborhoods of the central trajectory in terms of the \eigenvalues" of the second-order cone, and develop an analogue of HRVW/KSH/M direction, and establish O( p n log " 1 ), O(n log " 1 ) and O(n 3 log " 1 ) iteration-complexity bounds for short-step, semilong-step and long-step path-following algorithms, respectively, to reduce the duality gap by a factor of ". keywords: second-order cone, interior-point methods, polynomial complexity, primal-dual path-following methods.