2015/06/23 by Lei Yang, Ting Kei Pong, Yang, Lei +3 · 2 citations
Engineering · Mathematics · #Sparse and Compressive Sensing Techniques #Numerical methods in inverse problems #Photoacoustic and Ultrasonic Imaging
paper · pdf · doi:10.48550/arxiv.1506.07029
In this paper, we study a general optimization model, which covers a large class of existing models for many applications in imaging sciences. To solve the resulting possibly nonconvex, nonsmooth and non-Lipschitz optimization problem, we adapt the alternating direction method of multipliers (ADMM) with a general dual step-size to solve a reformulation that contains three blocks of variables, and analyze its convergence. We show that for any dual step-size less than the golden ratio, there exists a computable threshold such that if the penalty parameter is chosen above such a threshold and the sequence thus generated by our ADMM is bounded, then the cluster point of the sequence gives a stationary point of the nonconvex optimization problem. We achieve this via a potential function specifically constructed for our ADMM. Moreover, we establish the global convergence of the whole sequence if, in addition, this special potential function is a Kurdyka-Łojasiewicz function. Furthermore, we present a simple strategy for initializing the algorithm to guarantee boundedness of the sequence. Finally, we perform numerical experiments comparing our ADMM with the proximal alternating linearized minimization (PALM) proposed in [5] on the background/foreground extraction problem with real data. The numerical results show that our ADMM with a nontrivial dual step-size is efficient.