Convegno
“PRIMO WORKSHOP 2021”

Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well conditioned problems. In this talk, specialized variants of an interior point-proximal method of multipliers are proposed and analyzed for problems of this class. Computational experience on a variety of problems, namely, multi-period portfolio optimization, classification of data coming from functional Magnetic Resonance Imaging, restoration of images corrupted by Poisson noise, and classification via regularized logistic regression, provides substantial evidence that interior point methods, equipped with suitable linear algebra, can offer a noticeable advantage over first-order approaches. This is joint work with V. De Simone and M. Viola (University of Campania "L. Vanvitelli", Italy) and with J. Gondzio and S. Pougkakiotis (University of Edinburgh, UK).

Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretized, they lead to ill-conditioned linear systems, often of huge dimensions: regularization consists in replacing the original system by a nearby problem with better numerical properties, in order to find a meaningful approximation of its solution. After briefly surveying some standard regularization methods, both iterative (such as many Krylov methods) and direct (such as Tikhonov method), this talk will introduce a recent class hybrid projection methods, which merge an iterative and a direct approach to regularization. In particular, strategies for choosing the regularization parameter and the regularization matrix will be emphasized, eventually leading to the computation of approximate solutions of Tikhonov problems involving a regularization term expressed in a p-norm.