- A new class of large neighborhood path-following interior point algorithms for semidefinite optimization with $O(\sqrt{n}\log{\frac{{\rm Tr}(X^0S^0)}{\epsilon}})$ iteration complexity Yang Li(liy99gmail.com) Tamas Terlaky(terlakymcmaster.ca) Abstract: In this paper, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood $\N(\tau_1,\tau_2,\eta)$ 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 $(X^0,y^0,S^0)$ in $\N(\tau_1,\tau_2,\eta)$, the algorithm terminates in at most $O(\eta \sqrt{\kappa_{\infty}n}\log \frac{{\rm Tr}(X^0S^0)}{\epsilon})$ 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 when $\eta$ is chosen to be in the order of $\sqrt{n}$, our result coincides with the results for the classical large neighborhood IPMs. Keywords: interior point methods, large neighborhood, path-following algorithm, semidefinite optimization Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Citation: Technical Report 20078-04, Advanced Optimization Lab., Department of Computing and Software, Mcmaster University, Hamilton, Ontario, Canada. Download: [PDF]Entry Submitted: 07/02/2008Entry Accepted: 07/09/2008Entry Last Modified: 07/02/2008Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.