|Publication ahead of print|
|Published online||22 January 2018|
A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function∗
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria.
Corresponding author: firstname.lastname@example.org
Received: 3 November 2015
Revised: 30 June 2016
Accepted: 28 February 2017
We address the minimization of the sum of a proper, convex and lower semicontinuous function with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The trajectory generated by the dynamical system is proved to asymptotically converge to a critical point of the objective, provided a regularization of the latter satisfies the Kurdyka−Łojasiewicz property. Convergence rates for the trajectory in terms of the Łojasiewicz exponent of the regularized objective function are also provided.
Mathematics Subject Classification: 34G25 / 47J25 / 47H05 / 90C26 / 90C30 / 65K10
Key words: Dynamical systems / continuous forward-backward method / nonsmooth optimization / limiting subdifferential / Kurdyka−Łojasiewicz property
© EDP Sciences, SMAI 2018
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