This paper develops two implementations of Halpern-type proximal algorithms (HPA1 and HPA2) solving nonsmooth problems. We prove their convergence to the solutions of the problems under the new conditions. Applying this idea to the Basis Pursuit model in image/signal processing, a new Halpern-type proximal algorithm (HPA) for the model is obtained. Experiments show that the Halpern-type proximal algorithm has better descent property than the usual proximal algorithm (PA) for the Basis Pursuit model.
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