On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces

By Felipe Alvarez. SIAM Journal of Control and Optimization, 2000

334 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/96e83c6f0c78208d5fa0c8323632c0ca7d2f7358

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Citations in the corpus

A second-order gradient-like dissipative dynamical system with Hessian-driven damping.: Application to optimization and mechanics (2002, methodology): “This result, proved by Alvarez in [2], may be seen as an extension of the celebrated Bruck theorem for (SD) [16] to a second-order in time differential dynamical system; see also [3] for an implicit discrete proximal version of their result.”
On the second-order asymptotical regularization of linear ill-posed inverse problems (2018, positive): “an optimization viewpoint, establishing diﬀerent convergence results and identifying the circumstances under which the rate of convergence of HBF is better than the one of the ﬁrst order methods; see [1, 2, 3]. Numerical algorithms based on (1.4) for solving some special problems, e.g. inverse sourc”
Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity (2018, methodology): “Some of these results have been extended to inertial dynamics, see [2] for the heavy ball with friction, and [4] for an inertial version of Newton’s method.”
Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time (2009, background): “as the Cauchy property. Comments Theorem 4 in [46, Nevanlinna and Reich] shows that this result remains true if Xand X∗ are uniformly convex. In the same paper, the authors give a counterexample in C([0,1];R). See also [26, Bruck and Reich]. Compactness. Proposition 56 If ω[u(t)] ∩S 6= ∅ then u(t) c”
Convergence results of two-step inertial proximal point algorithm (2022, methodology): “As a result of the properties of (6), an implicit discretization method was proposed in [2, 3] as follows, given xn−1 and xn, the next point xn+1 is determined via 0 ∈ xn+1 − 2xn + xn−1 h(2) + β xn+1 − xn h + A(xn+1), (7)”
Inertial Douglas-Rachford splitting for monotone inclusion problems (2014, methodology): “Inertial proximal methods go back to [1, 3], where it has been noticed that the discretization of a differential system of second-order in time gives rise to a generalization of the classical proximal-point algorithm for finding the zeros of a maximally monotone operator (see [34]), nowadays called ”
Second-order differential equations with asymptotically small dissipation and piecewise flat potentials (2009, background): “When the map a is constant and positive, it is established in [1, 2] that the trajectories of (1.”
An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping (2001, methodology): “Even if we had existence of solutions, the asymptotic convergence in the continuous case would be an open question for a general A. Nevertheless, there are some positive results in this direction when A = ∂f (the proof is identical to [ 2, Theorem 3.1 ]) and when A is a cocoercive operator (cf.”
A note on the inertial proximal point method (2015, methodology): “Motivated by the properties of (4), an implicit discretization method was proposed in [2, 4].”
On damped second-order gradient systems (2014, background): “Out of the convex realm [45, 1], the seminal paper is probably [31].”
Fast optimization via inertial dynamics with closed-loop damping (2020, background): “We will recover and improve the results of Alvarez [7, Theorem 2.6], which concerns the case f convex, and the damped inertial equation ẍ(t) +A(ẋ(t)) +∇f(x(t)) = 0, where A : H → H is a positive definite self-adjoint linear operator, which is possibly anisotropic (see also [45]).”
ASYMPTOTIC BEHAVIOR OF SECOND-ORDER DISSIPATIVE EVOLUTION EQUATIONS COMBINING POTENTIAL WITH NON-POTENTIAL EFFECTS ∗ (2009, methodology): “In the potential case, A = ∇φ (corresponding to A = 0), taking advantage of the fact that, in the above result, no restrictive assumption is made on the potential operator ∇φ, we recover the asymptotic convergence result for the so-called heavy ball with friction dynamical system (Alvarez [2])”
Auxiliary Principle Technique for Equilibrium Problems (2004, neutral): “− un+1 − wn 2 . (21) Combining ( 20 ) and (21), we have”
Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space (2003, methodology): “IPPA was first considered in [1] for a (nonsmooth) conservative operator A = ∂f , the subdifferential of a closed, proper, and convex function f : H → R ∪ {∞}; weak convergence toward a minimizer of f holds under suitable conditions (see [1, Thm.”