Doctoral School Mathematics and Scientific Computing

Abstract: We describe a systematic procedure to analyze the convergence of degenerate preconditioned proximal point algorithms. We establish weak convergence results under mild assumptions that can be easily employed in the context of splitting methods for monotone inclusion and convex minimization problems. Moreover, we show that the degeneracy of the preconditioner allows for a reduction of the variables involved in the iteration updates. We show the strength of the proposed framework in the context of splitting algorithms, unifying the convergence analysis of several popular schemes, such as Douglas-Rachford, Chambolle-Pock, and Davis-Yin, which we analyze from a preconditioned proximal point perspective. We conclude by presenting a new sequential generalization of the Davis-Yin algorithm, along with numerical experiments that demonstrate its interest in the context of nonsmooth convex optimization.[hide abstract]