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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.

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