Full Nesterov-Todd Step Primal-Dual Interior-Point Methods for Second-Order Cone Optimization ∗ M. Zangiabadi †‡ G. Gu ‡ C. Roos ‡ † Department of Mathematical Science, Shahrekord University, P.O. Box 115, Shahrekord, Iran ‡ Department of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands e-mail: [M.Zangiabadi, G.Gu, C.Roos]@tudelft.nl Abstract After a brief introduction to Jordan algebras, we present a primal-dual interior-point algorithm for second-order conic optimization that uses full Nesterov-Todd-steps; no line searches are required. The number of iterations of the algorithm is O( √ N log(N/ε), where N stands for the number of second-order cones in the problem formulation and ε is the desired accuracy. The bound coincides with the currently best iteration bound for second- order conic optimization. We also generalize an infeasible interior-point method for linear optimization [26] to second-order conic optimization. As usual for infeasible interior-point methods the starting point depends on a positive number ζ . The algorithm either finds an ε-solution in at most O (N log(N/ε)) steps or determines that the primal-dual problem pair has no optimal solution with vanishing duality gap satisfying a condition in terms of ζ . 1 Introduction Second-order conic optimization (SOCO) problems are convex optimization problems that min- imize a linear objective function over the intersection of an affine linear manifold and the Carte- sian product of a finite number of second-order (or Lorentz or ice-cream) cones. Mathematically, a typical second-order cone in R n has the form L =  (x 1 ,x 2 ; ... ; x n ) ∈ R n : x 2 1 ≥ n  i=2 x 2 i ,x 1 ≥ 0  , (1) where n ≥ 2 is some natural number. * This paper was presented during the SIAM Conference on Optimization, May 10-13, 2008, in Boston, USA. 1