Proof Mining in Convex Optimization and related areas

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In this Project we aim at using proof-theoretic methods from logic for the extraction of new data (such as effective bounds, "proof mining") from prima facie noneffective proofs in convex optimization and related areas.In the course of this logic-based methodology, suitable forms of so-called proof interpretations have been developed by the applicant during the past decades and successfully applied in nonlinear analysis. In the previous 3 years of funding we applied this at large scale to problems in the area of convex optimization and - additionally - also carried out new case studies in neighbouring areas such as ergodic theory, approximation theory and Tauberian theory.During the next 3 years we will extend this approach to further problems in convex optimization including those which have a connection to current work in the area of machine learning. Here we will focus on proofs which make use of generalizations of the concept of "monotonicity" for set-valued operators which have recently been studied in convex optimization and are used in the context of machine learning. We also intend to analyze proofs which study abstract Cauchy problems given by accretive operators.The main goal of this project is the extraction of rates of asymptotic regularity, metastability (in the sense of T. Tao) and convergence of central iterative procedures whose convergence is shown using such generalized monotonicity or accretivity properties of set-valued operators but also the generalization of such results from the setting of Hilbert spaces to metric structures, such as CAT(0)-spaces, and more general Banach spaces.