Journal of Convex Analysis 25 (2018), No. 2, 371--388
Copyright Heldermann Verlag 2018
A Forward-Backward-Forward Differential Equation and its Asymptotic Properties
Sebastian Banert
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
sebastian.banert@univie.ac.at
Radu Ioan Bot
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
radu.bot@univie.ac.at
We approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with nonconstant stepsize function. Besides proving existence and uniqueness of strong global solutions for the differential equation under consideration, we show weak convergence of the generated trajectories and, under strong monotonicity assumptions, strong convergence with exponential rate. In the particular setting of minimizing the sum of a proper, convex and lower semicontinuous function with a smooth convex one, we provide a rate for the convergence of the objective function along the ergodic trajectory to its minimum value.
Keywords: Implicit dynamical system, continuous forward-backward-forward method, Lyapunov analysis, monotone inclusions, convex optimization.
MSC: 34G25, 47H05, 90C25
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