Abstract
In a Hilbert space setting, we introduce dynamical systems, which are linked to Newton and Levenberg-Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property, and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions. The Levenberg-Marquardt regularization term acts in an open loop way. As a byproduct of our study, we can take the regularization coefficient of bounded variation. These stability results are directly related to the study of numerical algorithms that combine forward-backward and Newton’s methods.
Highlights
Throughout this paper, H is a real Hilbert space with scalar product ., . and norm ·
As a guideline of our study, we use the Newton-like dynamic approach to solving monotone inclusions which was introduced in [7]. To adapt it to structured monotone inclusions and splitting methods, this study was developed in [7], [6], [2] and [5], where the operator is the sum of the subdifferential of a convex lower semicontinuous function, and the gradient of a convex differentiable function
We are going to consider some discrete and continuous Newton-like dynamics, which aim at solving structured monotone inclusions of the following type
Summary
Throughout this paper, H is a real Hilbert space with scalar product ., . and norm ·. We are concerned with the study of Newton-like continuous and discrete dynamics attached to solving the the structured minimization problem (P) min {φ(x) + ψ(x) : x ∈ H}. Let us stress the fact that, for each t > 0, the operators proxμ(t)φ : H −→ H, ∇φμ(t) : H −→ H are everywhere defined and Lipschitz continuous, which makes this system relevant to the Cauchy–Lipschitz theorem in the nonautonomous case, which naturally suggests good stability results of the solution of (4)-(6) with respect to the data. We consider the case where λ is locally absolutely continuous Note that it is important, for numerical reasons, to study the stability of the solution with respect to perturbations of the data, and in particular of λ which plays a crucial role in the regularization process. In Theorem 4.1 and Corollary 4.1, we prove the existence and uniqueness of a strong solution for (4)-(6), in the case where λ is a function with bounded variation
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