On the convergence of the proximal point algorithm for convex minimization
SIAM Journal on Control and Optimization
Mathematical control theory: deterministic finite dimensional systems (2nd ed.)
Mathematical control theory: deterministic finite dimensional systems (2nd ed.)
Central Paths, Generalized Proximal Point Methods, and Cauchy Trajectories in Riemannian Manifolds
SIAM Journal on Control and Optimization
Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Optimization
Kantorovich's theorem on Newton's method in Riemannian Manifolds
Journal of Complexity
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds
Foundations of Computational Mathematics
A Parallel Splitting Method for Coupled Monotone Inclusions
SIAM Journal on Control and Optimization
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We introduce nonautonomous 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.