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.

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Short Bio (15.07.2021):

The research interests of Professor Jacek Gondzio ( https://www.maths.ed.ac.uk/~gondzio/ ) include the theory and implementation of algorithms for very large-scale optimization. He is best known for his contributions in the area of interior point methods.

He was educated in Poland and since 1998 he has held a position at the School of Mathematics, the University of Edinburgh.

Prof Gondzio is an Editor of four leading optimization journals:

Computational Optimization and Applications, European Journal of Operational Research, Mathematical Programming Computation and Optimization Methods and Software.