Nonmonotone Barzilai-Borwein gradient algorithm for \(\ell_1\)-regularized nonsmooth minimization in compressive sensing
The authors propose, analyze and test a new practical algorithm to solve the separable nonsmooth minimization problem consisting of an \(\ell_1\)-norm regularized term and a continuously differentiable term. This type of problem mainly appears in signal/image processing, compressive sensing, machine learning, and linear inverse problem. However, the problem is challenging because of the non-smoothness of the regularization term. For various practical problems, either the computing efforts of the Hessian matrices are very expensive or the evaluation of the Hessian is difficult, the Hessian is not even available analytically. These challenges lead to the quasi-Newton method, which generates a series of Hessian approximation through the use of the gradient while maintaining a fast rate of convergence.
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