A New Class of Large Neighborhood Path-Following Interior Point Algorithms for Semidefinite Optimization with O(√n log (Tr(X0S0)/ε)) 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 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 to solve 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 (X0, y0, S0) in N (τ1, τ2, η), the algorithm terminates in at most O(η √ κ∞n log Tr(X0S0) ε ) iterations, where κ∞ is a parameter associated with the scaling matrix P and ε 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. Some preliminary numerical results are provided as well.