Employing recent outcomes in proximal theory outside the comfort zone

• Applicant: Professor Dr. Gert Wanka
• Subject Area: Mathematics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 281129760

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.

• DFG Programme: Research Grants
• International Connection: Austria
• Co-Investigator: Professor Dr. Radu Ioan Bot