Full Nesterov-Todd Step Primal-Dual Interior-Point Methods for Second-Order Cone Optimization∗
نویسندگان
چکیده
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 [C. Roos, A full-Newton step O(n) infeasible interior-point algorithm for linear optimization, 16(4) 2006, 1110-1136.] to second-order conic optimization. As usual for infeasible interiorpoint 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. ∗This paper was presented during the SIAM Conference on Optimization, May 10-13, 2008, in Boston, USA.
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