Investigations on monotonicity in general Banach spaces

With this project we intend to bring, by means of convex analysis, significant advances in the theory and applications of monotone operators defined on general Banach spaces. In particular we plan to provide improvements and extensions of classical results in this research field and new developments concerning the algorithmic approaches for solving monotone inclusion problems in Banach spaces. We propose a research plan structured in three main objectives, namely 1. investigations on monotonicity preserving operations in general Banach spaces; 2. inquiries on diagonal subdifferential operators via representative functions; 3. establishing new iterative methods for solving monotone inclusion problems in Banach spaces. In the first part of the project we plan to deal with monotone operators defined on general Banach spaces, in order to gain a deeper knowledge on their properties. We will investigate when do complex structures involving sums and compositions of monotone operators inherit the nice properties of their constituents. In particular we plan to thoroughly inspect some recent papers on Rockafellar`s conjecture, whose conclusions are highly contested in the community. The main tool in our investigations will be the representative functions, on which we expect new insights, too. A special and very intriguing class of monotone operators consists of the diagonal subdifferential operators. We plan to study them in this research project, too. Using the representative functions introduced earlier by the applicants, we expect to find new properties regarding domains, boundedness or surjectivity properties of this class of monotone operators. On the other hand, we plan to investigate which properties of the classical subdifferential are inherited by this class, too. The third major objective of this project concerns iterative methods for solving monotone inclusions in Banach spaces, i.e. problems of identifying a point in whose image set through a monotonicity preserving operation involving monotone operators lies a certain element. These contain as special cases many interesting classes of problems, in particular convex optimization ones. We expect to construct a new duality approach for monotone inclusions and to employ this for providing new primal-dual methods for solving monotone inclusions. Accelerations of these algorithms by means of inertial techniques will be investigated, too. The theoretical outcomes of this project are expected to bring new insights also in fields like convex optimization, equilibrium problems, variational inequalities, control theory or (partial) differential equations, while the algorithms will be employed on concrete applications arising from fields like finance mathematics, image processing or game theory.

As part of the project, new research results (algorithms for solving optimization problems, equivalent interpretations using dynamic systems and duality statements) were obtained.