A dynamical approach to convex minimization coupling approximation with the steepest descent method

H. Attouch, R. Cominetti

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Abstract

We study the asymptotic behavior of the solutions to evolution equations of the form 0 ∈u̇(t) + ∂f(u(t),ε(t)); u(0) = u0, where {f(·,ε):ε>0} is a family of strictly convex functions whose minimum is attained at a unique point x(ε). Assuming that x(ε) converges to a point x* as ε tends to 0, and depending on the behavior of the optimal trajectory x(ε), we derive sufficient conditions on the parametrization ε(t) which ensure that the solution u(t) of the evolution equation also converges to x* when t→ + ∞. The results are illustrated on three different penalty and viscosity-approximation methods for convex minimization.

Original languageEnglish
Pages (from-to)519-540
Number of pages22
JournalJournal of Differential Equations
Volume128
Issue number2
DOIs
StatePublished - 1 Jul 1996

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