A dynamic approach to a proximal-Newton method for monotone inclusions in Hilbert spaces, with complexity O(1/n^2)

In a Hilbert setting, we introduce a new dynamical system and associated algorithms for solving monotone inclusions by rapid methods. Given a maximal monotone operator A, the evolution is governed by the time dependent operator I -(I + λ(t) A)-1, where the positive control parameter λ(t) tends to infinity as t → + ∞. The tuning of λ (·) is done in a closed-loop way, by resolution of the algebraic equation λ norm(I + λ A)-1x -x=θ, where θ is a positive given constant. The existence and uniqueness of a strong global solution for the Cauchy problem follows from Cauchy-Lipschitz theorem. We prove the weak convergence of the trajectories to equilibria, and superlinear convergence under an error bound condition. When A =partial f is the subdifferential of a closed convex function f, we show a bigo(1/t2) convergence property of f(x(t)) to the infimal value of the problem. Then, we introduce proximal-like algorithms which can be obtained by time discretization of the continuous dynamic, and which share the same fast convergence properties. As distinctive features, we allow a relative error tolerance for the solution of the proximal subproblem similar to the ones proposed in ~citeSo-Sv1, So-Sv2, and a large step condition, as proposed in~citeMS1,MS2. For general convex minimization problems, the complexity is bigo(1/n2). In the regular case, we show the global quadratic convergence of an associated proximal-Newton method.