Polynomial Convergence of Primal-Dual Algorithms for Semidefinite Programming Based on the Monteiro and Zhang Family of Directions

This paper establishes the polynomialconvergence of the class of primal-dualfeasible interior-point algorithms for semideenite programming (SDP) based on Monteiro and Zhang family of search directions. In contrast to Monteiro and Zhang's work, no condition is imposed on the scaling matrix that determines the search direction. 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 SDP. Since Monteiro and Zhang family of directions includes the Alizadeh, Haeberly and Overton direction, we establish for the rst time the polynomial convergence of algorithms based on this search direction.