Topics in matrix analysis
SIAM Review
Self-scaled barriers and interior-point methods for convex programming
Mathematics of Operations Research
Mathematical Programming: Series A and B
SIAM Journal on Optimization
Primal-Dual Interior-Point Methods for Self-Scaled Cones
SIAM Journal on Optimization
SIAM Journal on Optimization
Primal--Dual Path-Following Algorithms for Semidefinite Programming
SIAM Journal on Optimization
On the Nesterov--Todd Direction in Semidefinite Programming
SIAM Journal on Optimization
SIAM Journal on Optimization
Journal of Computational and Applied Mathematics
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In this paper, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood $\mathcal{N}(\tau_1,\tau_2,\eta)$, and, as usual but with a small change, we make use of the scaled Newton equations for symmetric search directions. After defining the “positive part” and the “negative part” of a symmetric matrix, we recommend solving the Newton equation with its right-hand side replaced first by its positive part and then by its negative part, respectively. In such a way, we obtain a decomposition of the classical Newton direction and use different step lengths for each of them. Starting with a feasible point $(X^0,y^0,S^0)$ in $\mathcal{N}(\tau_1,\tau_2,\eta)$, the algorithm terminates in at most $O(\eta\sqrt{\kappa_{\infty}n}\log({Tr}(X^0S^0)/\epsilon))$ iterations, where $\kappa_{\infty}$ is a parameter associated with the scaling matrix $P$ and $\epsilon$ is the required precision. To our best knowledge, when the parameter $\eta$ is a constant, this is the first large neighborhood path-following interior point method (IPM) with the same complexity as small neighborhood path-following IPMs for semidefinite optimization that use the Nesterov-Todd direction. In the case where $\eta$ is chosen to be in the order of $\sqrt{n}$, our result coincides with the results for classical large neighborhood IPMs. Some preliminary numerical results also confirm the efficiency of the algorithm.