Interior Point Methods (IPMs) for linear and quadratic optimization have been very successful but occasionally they struggle with excessive memory requirements and high computational costs per iteration. The use of appropriately preconditioned iterative methods overcomes these drawbacks. In the first part of the talk, I will address the theoretical issues of applying the *inexact* Newton method in an IPM and a redesign of the method bearing two objectives in mind:\\ (a) avoiding explicit access to the problem data and allowing only matrix-vector products to be executed with the Hessian and Jacobian and its transpose; and\\ (b) allowing the method to work in a limited-memory regime. In the second part of the talk, I will comment on different preconditioners applicable in the IPM context and present the numerical results which demonstrate the practical performance of several of such preconditioners on challenging applications: - sparse approximation problems arising in compressed sensing, - PageRank (Google) problems. The computational experience reveals that the interior point method using inexact Newton directions outperforms the first order methods on many instances of these applications.

Jacek Gondzio @VideoLectures.net