Eintrag in der Universitätsbibliographie der TU Chemnitz
Volltext zugänglich unter
URN: urn:nbn:de:bsz:ch1-qucosa2-883961
Bitterlich, Sandy
Wanka, Gert (Prof. Dr. Dr. h.c.) (NUM) ; Szilárd Csaba, László (Prof. Dr.) ; Staudigl, Mathias (Assoc. Prof. Dr.) (Gutachter)
Numerical splitting methods for nonsmooth convex optimization problems
Kurzfassung in englisch
In this thesis, we develop and investigate numerical methods for solving nonsmooth convex optimization problems in real Hilbert spaces. We construct algorithms, such that they handle the terms in the objective function and constraints of the minimization problems separately, which makes these methods simpler to compute. In the first part of the thesis, we extend the well known AMA method from Tseng to the Proximal AMA algorithm by introducing variable metrics in the subproblems of the primal-dual algorithm. For a special choice of metrics, the subproblems become proximal steps. Thus, for objectives in a lot of important applications, such as signal and image processing, machine learning or statistics, the iteration process consists of expressions in closed form that are easy to calculate. In the further course of the thesis, we intensify the investigation on this algorithm by considering and studying a dynamical system. Through explicit time discretization of this system, we obtain Proximal AMA. We show the existence and uniqueness of strong global solutions of the dynamical system and prove that its trajectories converge to the primal-dual solution of the considered optimization problem. In the last part of this thesis, we minimize a sum of finitely many nonsmooth convex functions (each can be composed by a linear operator) over a nonempty, closed and convex set by smoothing these functions. We consider a stochastic algorithm in which we take gradient steps of the smoothed functions (which are proximal steps if we smooth by Moreau envelope), and use a mirror map to 'mirror'' the iterates onto the feasible set. In applications, we compare them to similar methods and discuss the advantages and practical usability of these new algorithms.
| Universität: | Technische Universität Chemnitz | |
| Institut: | Forschung Prof. Wanka | |
| Fakultät: | Fakultät für Mathematik | |
| Dokumentart: | Dissertation | |
| Betreuer: | Wanka, Gert Prof. Dr. Dr. h.c. (NUM) | |
| URL/URN: | https://nbn-resolving.org/urn:nbn:de:bsz:ch1-qucosa2-883961 | |
| SWD-Schlagwörter: | Konvexe Optimierung , Dualität , Dynamisches System , Konvergenz | |
| Freie Schlagwörter (Englisch): | Proximal AMA , dynamical system , Lyapunov analysis , Nesterov smoothing , mirror descent | |
| DDC-Sachgruppe: | Mathematik | |
| Sprache: | englisch | |
| Tag der mündlichen Prüfung | 10.05.2023 |
