## Monday, January 23, 2012

### The Bregman Iterative Procedure

Do not confuse this with the linearized Bregman method!

so says the Bregman Iterative Procedure site: where is featured this paper: Error Forgetting of Bregman Iteration. by  Wotao Yin and Stanley Osher. The abstract reads:

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 J(x)+ 12kAx−bkk22, where bk is iteratively updated. In practice, the subproblems can be solved at relatively low accuracy. Let w k denote the numerical error at iteration k. If all w k are sufficiently small so that Bregman iteration identifies the optimal face, then on the optimal face, Bregman iteration enjoys an interesting error-forgetting property: the distance between the current point ˉx k and the optimal solution set X is bounded by kwk+1− wkk, independent of the numerical errors at previous iterations. This property partially explains why the Bregman iterative procedure works well for sparse optimization and, in particular, l1-minimization. The error-forgetting property is unique to piece-wise linear functions (i.e., polyhedral functions) J(x), and the results of this article appears to new to the literature of the augmented Lagrangian method.
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