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Fixed-point continuation for $$\ell _1$$-minimization: methodology and convergence. (English) Zbl 1180.65076
This paper presents an approach for solving large-scale $$l_1$$ regularized convex minimization problems that arise for example in compressed sensing. The authors employ operator splitting and a path-following method to solve the optimization problem without requiring strict convexity. The corresponding fix-point iteration is discussed and its convergence is proven for a fixed value of the regularization parameter under standard assumptions. Subsequently, a continuation method, i.e., a path-following method is proposed, but not further analyzed. Some numerical results verify the presented algorithm. The paper gives a very comprehensive overview of the available results in this area and clearly connects the presented theory with the results already available in the literature.

##### MSC:
 65K05 Numerical mathematical programming methods 90C06 Large-scale problems in mathematical programming 90C25 Convex programming
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