We propose a general framework of iteratively reweighted 1 methods for solving p regularization problems. We prove that after some iteration k, the iterates generated by the proposed methods have the same support and sign as the limit points, and are bounded away from 0, so that the algorithm behaves like solving a smooth problem in the reduced space. As a result, the global convergence can be easily obtained and an update strategy for the smoothing parameter is proposed which can automatically terminate the updates for zero components. We show that p regularization problems are locally equivalent to a weighted 1 regularization problem and every optimal point corresponds to a Maximum A Posterior estimation for independently and non-identically distributed Laplace prior parameters. Numerical experiments exhibit the behaviors and the efficiency of our proposed methods.
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