A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function
Radu Ioan Boţ, Ernö Robert Csetnek · French digital mathematics library (Numdam) · 2018
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-Lojasiewicz property.Convergence rates for the trajectory in terms of the Lojasiewicz exponent of the regularized objective function are also provided.