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A forward-backward-forward differential equation and its asymptotic properties

Sebastian Banert(sebastian.banert***at***univie.ac.at)
Radu Ioan Bot(radu.bot***at***univie.ac.at)

Abstract: In this paper, 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 relaxation parameters and stepsizes of the resolvents. 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

Category 1: Other Topics (Dynamic Programming )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 03/26/2015
Entry Accepted: 03/26/2015
Entry Last Modified: 03/26/2015

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