A new class of large neighborhood path - following interior point algorithms for semidefinite optimization with O ( √ n log Tr ( X 0 S 0 ) 2 ) iteration complexity

In this paper, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood N (τ1, τ2, η) and, as usual, we utilize symmetric directions by scaling the Newton equation with special matrices. After defining the “positive part” and the “negative part” of a symmetric matrix, we solve the Newton equation with its right hand side replaced first by its positive part and then by its negative part, respectively. In this way, we obtain a decomposition of the usual Newton direction and use different step lengths for each of them. Starting with a feasible point (X0, y0, S0) in N (τ1, τ2, η), the algorithm terminates in at most O(η √ κ∞n log Tr(X0S0) 2 ) iterations, where κ∞ is a parameter associated with the scaling matrix P and 2 is the required precision. To our best knowledge, when the parameter η 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 when η is chosen to be in the order of √ n, our result coincides with the results for the classical large neighborhood IPMs.