Convex optimization via monotone operator theory

For convex optimization problems, the optimality conditions characterizing the set of optimal solutions can be written in many cases in form of monotone inclusion problems. The present project aims to provide improvements of classical theoretical results and some of their algorithmic realizations in convex nondifferentiable optimization, from the perspective of the monotone operator theory. We propose ourselves to attain three main objectives. The first one concerns the investigation of regularity conditions in convex optimization under the use of generalized interiority notions and based on some new separation results. For many applications the classical regularity conditions are not verified, though we expect to be able to overcome this situation with this new class of conditions, for which the quasi-relative interior plays a central role. In particular, applications to dynamic and market equilibrium problems will be considered. The second main objective of the project refers to the problem of the maximality of the sum of some complex structures in which maximally monotone operators are involved. This problem has a big relevance, for example, in the theory of partial differential equations and in the numerical approximation of the set of zeros of sums of maximally monotone operators, the particularization of the latter delivering corresponding numerical results for convex optimization problems. The connections between convex analysis and monotone operators, which have been in the last years intensively investigated, represent the main tool in this context. The third objective is with respect to the formulation of penalty-type algorithms for solving monotone inclusion problems and the study of their convergence properties. We aim to generalize the existing penalty-type schemes to more general inclusion problems, to establish and improve their convergence rates and to allow in the interative schemes summable erros in order to make them error tolerant. Penalty-type algorithms play an important role in constrained convex optimization, while we are in particular interested in their performances when solving problems arising in image processing.

Due to numerous applications in fields like signal and image processing, portfolio optimization, clusteranalysis, location theory, network communication, machine learning, etc., the design and investigation of numerical algorithms for solving structured convex optimization problems attracted in the last couple of years huge interest from the applied mathematics community. The main contributions of the project are in connection with the solving of highly structured optimization problems and their generalizations to monotone operator inclusions. The nonsmooth (set-valued) objects are evaluated in the algorithm by means of the proximal (resolvent) steps, while the smooth (single-valued) operators are processed through forward evaluations. Following the well-known paradigm that implicite/explicite discretization of first-order differential inclusions/equations leads to proximal/subgradient type algorithms, we investigated the obtained numerical schemes also from the continuous perspective, which allowed a more detailed study of the algorithms.