## FIXED-POINT CONTINUATION FOR ℓ1-MINIMIZATION: METHODOLOGY AND CONVERGENCE

Citations: | 44 - 9 self |

### BibTeX

@MISC{Hale_fixed-pointcontinuation,

author = {Elaine T. Hale and Wotao Yin and Yin Zhang},

title = { FIXED-POINT CONTINUATION FOR ℓ1-MINIMIZATION: METHODOLOGY AND CONVERGENCE},

year = {}

}

### OpenURL

### Abstract

We present a framework for solving large-scale ℓ1-regularized convex minimization problem: min �x�1 + µf(x). Our approach is based on two powerful algorithmic ideas: operator-splitting and continuation. Operator-splitting results in a fixed-point algorithm for any given scalar µ; continuation refers to approximately following the path traced by the optimal value of x as µ increases. In this paper, we study the structure of optimal solution sets; prove finite convergence for important quantities; and establish q-linear convergence rates for the fixed-point algorithm applied to problems with f(x) convex, but not necessarily strictly convex. The continuation framework, motivated by our convergence results, is demonstrated to facilitate the construction of practical algorithms.