Abstract
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.
Highlights
Let f : Rn → R ∪ {+∞} be a proper, convex, and lower semicontinuous function, and let g : Rn → R be a Fréchet differentiable function with β-Lipschitz continuous gradient, i.e., there exists β ≥ 0 such that ∇g(x)−∇g(y) ≤ β x − y for all x, y ∈ Rn
Radu Ioan Bot: Research partially supported by FWF (Austrian Science Fund), project I 2419-N32
Ernö Robert Csetnek: Research partially supported by FWF (Austrian Science Fund), project P 29809N32 and by an Advanced Fellowship STAR-UBB of Babes-Bolyai University, Cluj Napoca
Summary
Let f : Rn → R ∪ {+∞} be a proper, convex, and lower semicontinuous function, and let g : Rn → R be a (possibly nonconvex) Fréchet differentiable function with β-Lipschitz continuous gradient, i.e., there exists β ≥ 0 such that ∇g(x)−∇g(y) ≤ β x − y for all x, y ∈ Rn. Implicit dynamical systems related to both optimization problems and monotone inclusions have been considered in the literature by Attouch and Svaiter in [15], Attouch, Abbas and Svaiter in [2] and Attouch, Alvarez and Svaiter in [9] These investigations have been continued and extended in [21,22,23,24]. The aim of this manuscript is to study the asymptotic properties of the trajectory generated by the second-order dynamical system (2) under convexity assumptions for f and by allowing g to be nonconvex. The dynamical system investigated in this paper can be seen as a continuous counterpart of the inertial-type algorithms presented in [26] and [34]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
