Employing recent outcomes in proximal theory outside the comfort zone

The scope of this research project is twofold. It is meant to provide a framework for the continuation of the more than fifteen years long fruitful scientific collaboration of the applicants and, in the same time, to create an opportunity for three young promising graduate students in Mathematics to work on modern and very actual topics in the fields of optimization and applied functional analysis. The main scientific target of the project, which represents a follow-up to a recently successfully concluded research project of one of the applicants that was funded by the German Research Foundation, is to employ recent advances concerning some classical techniques used so far mainly for iteratively minimizing convex functions in Hilbert spaces to research fields lying outside their comfort zone. These methods evolve around the notion of proximality, which relies on evaluating a certain regularization of the addressed mathematical object. Due to its reliability, simplicity and accuracy, the proximal theory was successfully employed for solving nondifferentiable convex optimization problems and monotone inclusions with complex structures as well, proving a strong positive impact on the treatment of real-life applications with high-dimensional data. The research themes we address in this project are structured within five objectives. They range from the employment of the paradigm of proximality in broader frameworks like considering generalized distances, working in more general underlying spaces and addressing the direct solving of multiobjective optimization problems to the approach of monotone inclusions problems via implicit first- and second-order dynamical systems. The expected results should have impact beyond the corresponding research areas both in mathematical fields like ordinary differential equations, partial differential equations, optimal control, functional analysis, game theory, equilibrium problems and optimal transport theory, and in real-life problems arising in optimal location selection, image processing, machine learning, quantification of risk, network communication and video processing.

Many problems which are nowadays relevant for the scientific and technological evolution of the society lead to high-dimensional optimization problems with complex structures for the solving of which specific numerical algorithms have to be designed. These include problems arising in optimal location selection, signal processing, machine learning, quantification of risk, network communication, game theory, and energy markets. The main focus of this project was the development of deterministic but also stochastic numerical methods for nonsmooth convex and nonconvex optimization problems, the theoretical investigation of their convergence properties and their validation by means of numerical experiments. The theoretical considerations were related to questions such as under which assumptions the provided algorithms converge, possibly with provable convergence rates, how to induce an improved convergence behavior, or how to extend the deterministic methods to a stochastic setting. In the framework of the project different mathematical approaches have been employed: the theory of maximally monotone operators and the solving of monotone inclusions; the asymptotic analysis of first- and second-order dynamical systems approaching the solutions/critical set of an optimization problem; the theory of KL functions for guaranteeing global convergence of algorithms for nonconvex optimization problems. The main application field was machine learning and data science, for which new numerical methods for the training of deep neural networks, in particular of generative adversarial networks were provided. Some of the developed numerical methods found application and implementation in collaborations with two Austrian industrial partners from the automotive and the energy sector. With the latter follow-up projects are envisaged. The project was carried out in collaboration with the research group "Approximation Theory" of the Chemnitz University of Technology, which was funded by the German Research Foundation. Two PhD students and one PostDoc were employed at the University of Vienna, while one PhD student has been employed at the Chemnitz University of Technology. The two PhD students on the Austrian side have already successfully graduated, while the one on the German side will finish her PhD studies soon. The PostDoc successfully applied for a follow-up research project. A total of 19 publications resulted from the research project, the large majority of them published in top-ranked optimization and applied mathematics journals. Many of these accumulated a large number of citations in a short time. The obtained results have been presented by the principal investigator and the project members at important international conferences and workshops, in part as keynote presentations. A final workshop with the participation of 20 members and collaborators of the research group of the principal investigator took place and provided an opportunity for presenting the results obtained in the framework of the project.