In this talk we consider a family of optimization problems which arise in the field of signal reconstruction, i.e. L1 and Total-Variation (TV) regularized Least-Squares (LS), L1-Analysis and combinations. There has been a considerable effort for the development of first-order algorithms for L1 and TV regularized LS. State-of-the-art implementations such as SPGL1 and TwIST can solve large-scale problems in few seconds on a PC. However, the broader family of problems studied in this talk challenge these methods. We solve this family of problems in two steps. First, appropriate smoothing of the problems is applied, second, a class of Newton-type algorithms is employed, embedded in a continuation framework for further acceleration. Moreover, we present perturbation analysis of the optimal solution obtained by the smoothed problem as a function of a smoothing parameter.
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