Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces

被引：80

作者：

Abbas, B. ^{[1
]}

Attouch, H. ^{[1
]}

Svaiter, Benar F. ^{[2
]}

机构：

[1] Univ Montpellier 2, UMR CNRS 5149 I3M, F-34095 Montpellier, France

[2] IMPA, BR-22460320 Rio De Janeiro, Brazil

来源：

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS | 2014年 / 161卷 / 02期

关键词：

Monotone inclusions; Newton method; Levenberg-Marquardt regularization; Dissipative dynamical systems; Lyapunov analysis; Weak asymptotic convergence; Forward-backward algorithms; Gradient-projection methods; CONVERGENCE; EQUATIONS; OPERATORS;

D O I：

10.1007/s10957-013-0414-5

中图分类号：

C93 [管理学]; O22 [运筹学];

学科分类号：

070105 ; 12 ; 1201 ; 1202 ; 120202 ;

摘要：

In a Hilbert space setting we introduce dynamical systems, which are linked to Newton and Levenberg-Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M=A+B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton's method and forward-backward methods for solving structured monotone inclusions.

引用

页码：331 / 360

页数：30