An Inertial Proximal Algorithm with Dry Friction: Finite Convergence Results期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

首页 | 本学科首页

官方微博 | 高级检索

按检索

An Inertial Proximal Algorithm with Dry Friction: Finite Convergence Results

Authors:

Bruno Baji and Alexandre Cabot

Institution:

(1) Laboratoire LACO, Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges, Cedex, France

Abstract:

Let H be a Hilbert space and A, B: H ⇉ H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm:

${{\left( {x_{{n + 2}} - 2x_{{n + 1}} + x_{n} } \right)}} \mathord{\left/ {\vphantom {{{\left( {x_{{n + 2}} - 2x_{{n + 1}} + x_{n} } \right)}} {\lambda ^{2}_{n} }}} \right. \kern-\nulldelimiterspace} {\lambda ^{2}_{n} } + A{\left( {{{\left( {x_{{n + 2}} - x_{{n + 1}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {x_{{n + 2}} - x_{{n + 1}} } \right)}} {\lambda _{n} }}} \right. \kern-\nulldelimiterspace} {\lambda _{n} }} \right)} + B{\left( {x_{{n + 2}} } \right)} \ni 0,\,\,\,\,\,\,\,\,\,\,\,{\left( {\user1{\mathcal{A}}} \right)}$

where (λ_n) is a sequence of positive steps. Algorithm ${\left( {\user1{\mathcal{A}}} \right)}$ may be viewed as the discretized equation of a nonlinear oscillator subject to friction. We prove that, if 0 ∈ int (A(0)) (condition of dry friction), then the sequence (x _n) generated by ${\left( {\user1{\mathcal{A}}} \right)}$ is strongly convergent and its limit x _∞ satisfies 0 ∈ A(0) + B(x _∞). We show that, under a general condition, the limit x _∞ is achieved in a finite number of iterations. When this condition is not satisfied, we prove in a rather large setting that the convergence rate is at least geometrical.

Keywords:

Mathematics Subject Classifications (2000)" target="_blank">Mathematics Subject Classifications (2000) 65K10 49M25 70F40

本文献已被 SpringerLink 等数据库收录！

设为首页 | 免责声明 | 关于勤云 | 加入收藏