Polynomial Convergence of Infeasible-Interior-Point Methods over Symmetric Cones

Abstract

We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of search directions used in Schmieta and Alizadeh [Math. Program., 96 (2003), pp. 409-438]. Monteiro and Zhang [Math. Program., 81 (1998), pp. 281-299] introduced this family of directions when analyzing semidefinite programs. These conic programs include linear and semidefinite programs. This extends the work of Rangarajan and Todd [Tech. rep. 1388, School of OR & IE, Cornell University, Ithaca, NY, 2003], which established convergence of infeasible-interior-point methods for self-scaled conic programs using the NT direction. Our work is built on earlier analyses by Faybusovich [J. Comput. Appl. Math., 86 (1997), pp. 149-175] and Schmieta and Alizadeh [Math. Program., 96 (2003), pp. 409-438]. Of independent interest, we provide a constructive proof of Lyapunov lemma in the Jordan algebraic se...