A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function

• Published in: Applied mathematics and computation Vol. 394; p. 125822
• Main Authors: Boţ, Radu Ioan, Kanzler, Laura
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.04.2021; Elsevier
• Subjects: Asymptotic analysis; Block-coordinate minimization; Forward-backward dynamical system; Kurdyka-Łojasiewicz property; Mathematics; Kurdyka-Łojasiewicz property; 49J52; 90C56; 90C26; Block-coordinate minimization; Asymptotic analysis; 34G25; 37N40; Forward-backward dynamical system; block-coordinate minimization forward-backward dynamical system asymptotic analysis Kurdyka-Łojasiewicz property AMS subject classification. 34G25 37N40 49J52 90C26 90C56; forward-backward dynamical system; Kurdyka-Łojasiewicz property AMS subject classification. 34G25; asymptotic analysis; block-coordinate minimization
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2020.125822

In this paper we aim to minimize the sum of two nonsmooth (possibly also nonconvex) functions in separate variables connected by a smooth coupling function. To tackle this problem we choose a continuous forward-backward approach and introduce a dynamical system which is formulated by means of the partial gradients of the smooth coupling function and the proximal point operator of the two nonsmooth functions. Moreover, we consider variable rates of implicity of the resulting system. We discuss the existence and uniqueness of a solution and carry out the asymptotic analysis of its convergence behaviour to a critical point of the optimization problem, when a regularization of the objective function fulfills the Kurdyka-Łojasiewicz property. We further provide convergence rates for the solution trajectory in terms of the Łojasiewicz exponent. We conclude this work with numerical simulations which confirm and validate the analytical results.