期刊
JOURNAL OF SCIENTIFIC COMPUTING
卷 54, 期 2-3, 页码 684-695出版社
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-012-9616-5
关键词
Bregman iteration; Error forgetting; Sparse optimization; l(1) minimization; Piece-wise linear function; Polyhedral function; Augmented Lagrangian method; Method of multipliers
资金
- NSF
- ONR
- DARPA
- AFOSR
- ARL
- ARO
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0748839] Funding Source: National Science Foundation
- Div Of Electrical, Commun & Cyber Sys
- Directorate For Engineering [1028790] Funding Source: National Science Foundation
This short article analyzes an interesting property of the Bregman iterative procedure, which is equivalent to the augmented Lagrangian method, for minimizing a convex piece-wise linear function J(x) subject to linear constraints Ax=b. The procedure obtains its solution by solving a sequence of unconstrained subproblems of minimizing , where b (k) is iteratively updated. In practice, the subproblem at each iteration is solved at a relatively low accuracy. Let w (k) denote the error introduced by early stopping a subproblem solver at iteration k. We show that if all w (k) are sufficiently small so that Bregman iteration enters the optimal face, then while on the optimal face, Bregman iteration enjoys an interesting error-forgetting property: the distance between the current point and the optimal solution set X (au) is bounded by ayenw (k+1)-w (k) ayen, independent of the previous errors w (k-1),w (k-2),aEuro broken vertical bar,w (1). This property partially explains why the Bregman iterative procedure works well for sparse optimization and, in particular, for a (1)-minimization. The error-forgetting property is unique to J(x) that is a piece-wise linear function (also known as a polyhedral function), and the results of this article appear to be new to the literature of the augmented Lagrangian method.
作者
我是这篇论文的作者
点击您的名字以认领此论文并将其添加到您的个人资料中。
推荐
暂无数据