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

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Renato D.C. Monteiro and Takashi T. Tsuchiya

This paper establishes the polynomial convergence of a new class of
(feasible) primal-dual interior-point path following algorithms for
semidefinite programming (SDP) whose search directions are obtained
by applying Newton method to the symmetric central path equation
$$(P^T X P)^{1/2}(P^{-1} S P^{-T}) ( P^T X P)^{1/2} - \mu I =0,$$
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 $O(n^{3/2}L)$ 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.
Report Memorandum No. 627, The Institute of Statistical
Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106,
JAPAN.

Contact: monteiro@isye.gatech.edu