Abstract: The aim of this lecture is to introduce the main tools and techniques specific to variational analysis which are required to solve nonsmooth optimization and monotone inclusion problems. The main theoretical aspects to be presented are: elements of convex analysis,
duality theory for convex optimization problems, regularity and optimality conditions, elements of monotone operator theory, proximal/resolvent operators.

The main focus will be on splitting methods for solving monotone inclusions/convex optimization
problems, including the following methods: forward-backward (proximal-gradient), Douglas-Rachford, forward-backward-forward (proximal-gradient-proximal).

The algorithms presented can be tested in several applied fields, like image processing, image inpainting, image segmentation, machine learning in connection to support vector classification and support vector regression, location theory, clustering, network communication, etc.

Key words: convex analysis, subdifferential calculus, monotone operators, proximal/splitting methods