Nonsmooth optimization problems: splitting and dynamics

Many real-life problems can be modeled as structured optimization problems, involving both nonsmooth and smooth data. The aim of this project is to deal with nonsmooth convex and nonconvex optimization problems having complex structures, by making use of splitting algorithms. The main feature of these algorithms is that they fully decompose all the objects from the model, which is a very useful and important aspect from the point of view of providing numerical implementations. The project has two main directions: investigations of structured nonsmooth optimization problems by means of proximal-type splitting algorithms and the analysis of their continuous counterparts, which are generically first order and second order differential equations attached to the optimization problem. It is well known in the literature that a well understanding of the continuous case delivers more insights into their time discretization versions. We start with investigations on the speed of convergence and acceleration techniques for splitting algorithms designed to solve highly structured convex and nonconvex and nonsmooth optimization problems. The main motivation is to go beyond the setting of acceleration techniques which are known up to now in the literature. More precisely, motivated by concrete applications in real-life problems, plenty of them can be modeled as optimization problems involving compositions with linear operators. For this kind of problems, there is a lack in the literature in what concerns the convergence rates which have been established for the fast gradient and FISTA method, respectively. We aim to investigate this issue for optimization problems with a more intricate nature, enlarging in this way the area of applications considered until now in the literature. Furthermore, we intend to investigate the iterative schemes also from the continuous perspective, by paying attention to differential inclusions/equations involving time dependent operators. One of the main applications of the dynamical systems governed by time dependent monotone operators are the sweeping processes, originally considered by Moreau in the study of evolution problems from unilateral mechanics. The applications we expect range from image processing, with a particular emphasis on the solving of real-life problems dealing with image denoising, image deblurring, image inpainting and image segmentation, to machine learning in connection with support vector classification and support vector regression. Other fields that we have in mind for numerical experiments are optimal portfolio selection, signal processing, decomposition of video streams, clustering and network communication.

Many real-life problems can be modeled as structured optimization problems, involving both nonsmooth and smooth data. The aim of this project is to deal with nonsmooth convex and nonconvex optimization problems having complex structures, by making use of splitting algorithms. The main feature of these algorithms is that they fully decompose all the objects from the model, which is a very useful and important aspect from the point of view of providing numerical implementations. The project has two main directions: investigations of structured nonsmooth optimization problems by means of proximal-type splitting algorithms and the analysis of their continuous counterparts, which are generically first order and second order differential equations attached to the optimization problem. It is well known in the literature that a well understanding of the continuous case delivers more insights into their time discretization versions. We report investigations on the speed of convergence and acceleration techniques for splitting algorithms designed to solve highly structured convex and nonconvex and nonsmooth optimization problems. The main motivation is to go beyond the setting of acceleration techniques which are known up to now in the literature. More precisely, motivated by concrete applications in real-life problems, plenty of them can be modeled as optimization problems involving compositions with linear operators. For this kind of problems, there is a lack in the literature in what concerns the convergence rates which have been established for the fast gradient and FISTA method, respectively. Furthermore, we investigated the iterative schemes also from the continuous perspective, by paying attention on differential inclusions/equations in the context of optimization problems/monotone inclusions. Fast optimization methods in the sense of Nesterov play a central role in the investigations, as well as the discretized counterparts of the dynamical systems considered. The applications range from image processing, with a particular emphasis on the solving of real-life problems dealing with image denoising, image deblurring, image inpainting and image segmentation, to machine learning in connection with support vector classification and support vector regression. Other fields that can be considered for numerical experiments are optimal portfolio selection, signal processing, decomposition of video streams, clustering and network communication.