Geodesic
A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 463-477

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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.

DOI : 10.1051/cocv/2017020
Classification : 34G25, 47J25, 47H05, 90C26, 90C30, 65K10
Keywords: Dynamical systems, continuous forward-backward method, nonsmooth optimization, limiting subdifferential, Kurdyka−Łojasiewicz property

Ioan Boţ, Radu 1 ; Robert Csetnek, Ernö 1

1 
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     author = {Ioan Bo\c{t}, Radu and Robert Csetnek, Ern\"o},
     title = {A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {463--477},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {2},
     year = {2018},
     doi = {10.1051/cocv/2017020},
     zbl = {1428.90128},
     mrnumber = {3816401},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017020/}
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Ioan Boţ, Radu; Robert Csetnek, Ernö. A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 463-477. doi: 10.1051/cocv/2017020

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