Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Radu Ioan Boţ

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

radu.bot@univie.ac.at

Keywords

  • Implicit dynamical system
  • continuous forward-backward-forward method
  • Lyapunov analysis
  • monotone inclusions
  • convex optimization

Mathematics Subject Classification

34G25, 47H05, 90C25

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Suggested citation

S. Banert, R. I. Boţ. “A Forward-Backward-Forward Differential Equation and its Asymptotic Properties.” Journal of Convex Analysis 25 (2018), No. 2, 371–388. https://doi.org/10.68381/jca25026

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press.