|Affiliation:||University of Edinburgh|
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Having briefly introduced the key ideas which make interior point methods (IPMs) such a powerful optimization approach, I shall focus on a solution of the Newton systems and in particular on the use of iterative (Krylov-subspace) techniques to perform this task.
The Newton systems arising in IPMs are inherently ill-conditioned and preconditioning is a must to make iterative methods work.
A re-design of IPMs to enable the use of iterative techniques provides a completely new perspective on these methods.
I will address both theoretical and practical aspects of it.Such a re-design opens interesting opportunities which include:
(i) relaxing the rigid structure of IPMs, and
(ii) developing matrix-free variants of the method.
Several such developments will be briefly discussed including some astonishing extensions of IPMs tuned to inverse problems and various formulations of sparse approximation problems.