Weak Versus Strong Convergence of a Regularized Newton Dynamic for Maximal Monotone Operators
| Autor: | Hedy Attouch, Jean-Bernard Baillon |
|---|---|
| Přispěvatelé: | Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
General Mathematics
0211 other engineering and technologies Levenberg-Marquardt reg Monotonic function 02 engineering and technology Lambda Newton-like continuous dynamic 01 natural sciences Combinatorics symbols.namesake Operator (computer programming) 0101 mathematics [MATH]Mathematics [math] weak versus strong asymptotic convergence Mathematics Key-words maximal monotone operators 021103 operations research ularization 010102 general mathematics Hilbert space Solution set Monotone polygon symbols [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] Convex function Baillon's counterexample Counterexample |
| Zdroj: | Vietnam Journal of Mathematics Vietnam Journal of Mathematics, Springer, 2018, 46 (1), pp.177-195. ⟨10.1007/s10013-017-0267-6⟩ |
| ISSN: | 2305-221X 2305-2228 |
| DOI: | 10.1007/s10013-017-0267-6⟩ |
| Popis: | In a Hilbert space $\mathcal {H}$ , given $A:\mathcal {H} \rightrightarrows \mathcal {H}$ a general maximal monotone operator whose solution set is assumed to be non-empty, and λ(⋅) a time-dependent positive regularization parameter, we analyze, when t → + ∞, the weak versus strong convergence properties of the trajectories of the Regularized Newton dynamic $$\text{(RN)} \quad \left\{ \begin{array}{l} v(t)\in A(x(t)),\\ \lambda (t) \dot x(t) + \dot v(t) + v(t) =0. \end{array}\right. $$ The term $\lambda (t) \dot x(t)$ acts as a Levenberg–Marquardt regularization of the continuous Newton dynamic associated with A, which makes (RN) a well-posed system. The coefficient λ(t) is allowed to tend to zero as t → + ∞, which makes (RN) asymptotically close to the Newton continuous dynamic. As a striking property, when λ(t) does not converge too rapidly to zero as t → + ∞ (with λ(t) = e −t as the critical size), Attouch and Svaiter showed that each trajectory generated by (RN) converges weakly to a zero of A. By adapting Baillon’s counterexample, we show a situation where A is the gradient of a smooth convex function, and there is a trajectory of the corresponding system (RN) that does not converge strongly. On the positive side, under certain particular assumptions about the operator A, or on the regularization parameter λ(⋅), we show the strong convergence when t → + ∞ of the (RN) trajectories. |
| Databáze: | OpenAIRE |
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