Second Order Forward-Backward Dynamical Systems For Monotone Inclusion Problems

Abstract

We begin by considering second order dynamical systems of the fromẍ(t)+γ(t)ẋ(t)+ λ(t)B(x(t)) = 0, where B : H → H is a cocoercive operator defined on a real Hilbert space H, λ : [0, +∞) → [0, +∞) is a relaxation function and γ : [0, +∞) → [0, +∞) a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator B. The framework allows to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one. This captures as particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Furthermore, we prove that when B is the gradient of a smooth convex function the value of the latter converges along the ergodic trajectory to its minimal value with a rate of O(1/t).

On the second-order asymptotical regularization of linear ill-posed inverse problems

“…• x(·),ẋ(·) : [0, +∞) → X are locally absolutely continuous [22], •ẍ(t) + ηẋ(t) + A * Ax(t) = A * y δ holds for almost every t ∈ [0, +∞).…”

“…The proof of Lemma 2 follows as a special case for f (x) = 1 2 Ax − y 2 in [22], and it is given in the Appendix A.2. The rate Ax(t) − y = o(t −1/2 ) as t → ∞ given in Lemma 2 for the second order evolution equation (4) should be compared with the corresponding result for the first order method, i.e.…”

Approaching nonsmooth nonconvex minimization through second-order proximal-gradient dynamical systems

“…Implicit dynamical systems related to both optimization problems and monotone inclusions have been considered in the literature also by Attouch and Svaiter in [15], Attouch, Abbas and Svaiter in [2] and Attouch, Alvarez and Svaiter in [9]. These investigations have been continued and extended in [21][22][23][24].…”

Second-order dynamical systems with penalty terms associated to monotone inclusions

“…where u 0 , v 0 ∈ H and γ, λ, β : [0, +∞) −→ (0, +∞). Dynamical systems governed by resolvents of maximally monotone operators have been considered in [1,2], and then further developed in [17,19].…”

“…For B = 0 and λ is constant, the differential equation (6) becomes the second order forward-backward dynamical system investigated in [19] in relation to the monotone inclusion problem…”