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A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function

Radu Ioan Bot(radu.bot***at***univie.ac.at)
Ernö Robert Csetnek(ernoe.robert.csetnek***at***univie.ac.at)

Abstract: We address the minimization of the sum of a proper, convex and lower semicontinuous 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-\L{}ojasiewicz property. Convergence rates for the trajectory in terms of the \L{}ojasiewicz exponent of the regularized objective function are also provided.

Keywords: dynamical systems, continuous forward-backward method, nonsmooth optimization, limiting subdifferential, Kurdyka-\L{}ojasiewicz property

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Citation:

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Entry Submitted: 07/06/2015
Entry Accepted: 07/06/2015
Entry Last Modified: 07/06/2015

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