A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

Abstract

We introduce non-autonomous continuous dynamical systems which are linked to the Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on the Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant to the Cauchy-Lipschitz theorem. By using Lyapunov methods, we prove that their trajectories converge weakly to equilibria. Time discretization of these dynamics gives algorithms providing new insight into Newton's method for solving monotone inclusions.2010 Mathematics Subject Classification: 34G25, 47J25, 47J30, 47J35, 49M15, 49M37, 65K15, 90C25, 90C53.

Backward–forward algorithms for structured monotone inclusions in Hilbert spaces

“…The alternative, a backward-forward algorithm, has not been explored. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators proposed in [11], and thereafter extended to the case of structured monotone operators in [2]. A semi-implicit discretization of the dynamical system studied in [9] (different from the one considered in [10]) produces this type of methods as well.…”

Approaching Nonsmooth Nonconvex Optimization Problems Through First Order Dynamical Systems with Hidden Acceleration and Hessian Driven Damping Terms

“…By applying Lemma 4, it follows that lim t→+∞ẋ (t) = 0. Moreover, from (12) we get thatÿ exists and y ∈ L 2 ([0, +∞); R n ) due to (21). The same arguments are used in order to conclude lim t→+∞ẏ (t) = 0.…”

“…Proximal-gradient dynamical systems, which are generalizations of (5), have been recently considered by Abbas and Attouch in [1,Section 5.2] in the full convex setting. Implicit dynamical systems related to both optimization problems and monotone inclusions have been considered in the literature also by Attouch and Svaiter in [21], Attouch, Abbas and Svaiter in [2] and Attouch, Alvarez and Svaiter in [11]. These investigations have been continued and extended in [22,32,[34][35][36].…”

“…see, for instance,[2,21]) A function x : [0, +∞) → R n is said to be locally absolutely continuous, if it absolutely continuous on every interval [0, T ], where T > 0.Remark 2 (a) An absolutely continuous function is differentiable almost everywhere, its derivative coincides with its distributional derivative almost everywhere and one can recover the function from its derivativeẋ = y by integration. (b) If x : [0, T ] → R n is absolutely continuous for T > 0 and B : R n → R n is L-Lipschitz continuous for L ≥ 0, then the function z = B • x is absolutely continuous, too.…”

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

“…This nonlinear oscillator with damping is, in case n = 2, a simplified version of the differential system describing the motion of a heavy ball that rolls over the graph of g and keeps rolling under its own inertia until friction stops it at a critical point of g (see [14]). 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].…”