Polynomial Convergence of a New Family of Primal-Dual Algorithms for Semidefinite Programming

This paper establishes the polynomial convergence of a new class of primal-dual interior-point path-following feasible algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton's method to the symmetric central path equation $$ (P X P^T)^{1/2}(P^{-T} S P^{-1}) ( P X P^T)^{1/2} - \mu I =0, $$ where P is a nonsingular matrix. Specifically, we show that the short-step path-following algorithm based on the Frobenius norm neighborhood and the semilong-step path-following algorithm based on the operator 2-norm neighborhood have $O(\sqrt{n}L)$ and O(nL) iteration-complexity bounds, respectively. When P = I, this yields the first polynomially convergent semilong-step algorithm based on a pure Newton direction. Restricting the scaling matrix P at each iteration to a certain subset of nonsingular matrices, we are able to establish an O(n3/2L) iteration complexity for the long-step path-following method. The resulting subclass of search directions contains both the Nesterov--Todd direction and the Helmberg--Rendl--Vanderbei--Wolkowicz/Kojima--Shindoh--Hara/Monteiro direction.