Lagrangian methods for composite optimization

Lagrangian-based methods have been on the market for over 50 years. These methods are robust and often can handle optimization problems with complex geometries through efficient computational steps. The last decade of research have generated a large volume of literature on various practical and theoretical aspects of many Lagrangian-based algorithms. This chapter reviews the basic elements of Lagrangian-based methods for composite minimization in the convex and nonconvex setting. In the convex case, the focus is on global rate of convergence results, which are derived here through a novel approach and very simple proof technique. In the much harder nonconvex case, we survey a very recent methodology which allows to establish global pointwise convergence results for a broad class of genuine nonlinear composite semialgebraic problems.