Polynomial convergence of primal–dual algorithms for the second-order cone program based on the Monteiro–Zhang family of directions

In this paper we study primal–dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro–Zhang (MZ) family for semidefinite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor–corrector algorithm of Mizuno et al., carry over to the context of SOCP, that is they have an O(√(n) log ϵ^–1) iteration-complexity to reduce the duality gap by a factor of ϵ, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh, Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal–dual algorithms for SOCP based on this search direction.