1,028 research outputs found
Well-posedness via Monotonicity. An Overview
The idea of monotonicity (or positive-definiteness in the linear case) is
shown to be the central theme of the solution theories associated with problems
of mathematical physics. A "grand unified" setting is surveyed covering a
comprehensive class of such problems. We elaborate the applicability of our
scheme with a number examples. A brief discussion of stability and
homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte
Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator
In a Hilbert framework, we introduce continuous and discrete dynamical
systems which aim at solving inclusions governed by structured monotone
operators , where is the subdifferential of a
convex lower semicontinuous function , and is a monotone cocoercive
operator. We first consider the extension to this setting of the regularized
Newton dynamic with two potentials. Then, we revisit some related dynamical
systems, namely the semigroup of contractions generated by , and the
continuous gradient projection dynamic. By a Lyapunov analysis, we show the
convergence properties of the orbits of these systems.
The time discretization of these dynamics gives various forward-backward
splitting methods (some new) for solving structured monotone inclusions
involving non-potential terms. The convergence of these algorithms is obtained
under classical step size limitation. Perspectives are given in the field of
numerical splitting methods for optimization, and multi-criteria decision
processes.Comment: 25 page
Monotone Inclusions, Acceleration and Closed-Loop Control
We propose and analyze a new dynamical system with a closed-loop control law
in a Hilbert space , aiming to shed light on the acceleration
phenomenon for \textit{monotone inclusion} problems, which unifies a broad
class of optimization, saddle point and variational inequality (VI) problems
under a single framework. Given
that is maximal monotone, we propose a closed-loop control system that is
governed by the operator , where a feedback law
is tuned by the resolution of the algebraic equation
for some
. Our first contribution is to prove the existence and uniqueness
of a global solution via the Cauchy-Lipschitz theorem. We present a simple
Lyapunov function for establishing the weak convergence of trajectories via the
Opial lemma and strong convergence results under additional conditions. We then
prove a global ergodic convergence rate of in terms of a gap
function and a global pointwise convergence rate of in terms of a
residue function. Local linear convergence is established in terms of a
distance function under an error bound condition. Further, we provide an
algorithmic framework based on the implicit discretization of our system in a
Euclidean setting, generalizing the large-step HPE framework. Although the
discrete-time analysis is a simplification and generalization of existing
analyses for a bounded domain, it is largely motivated by the above
continuous-time analysis, illustrating the fundamental role that the
closed-loop control plays in acceleration in monotone inclusion. A highlight of
our analysis is a new result concerning -order tensor algorithms for
monotone inclusion problems, complementing the recent analysis for saddle point
and VI problems.Comment: Accepted by Mathematics of Operations Research; 42 Page
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