In this paper, an alternating direction nonmonotone approximate Newton algorithm (ADNAN) based on nonmonotone line search is developed for solving inverse problems. It is shown that ADNAN converges to a solution of the inverse problems and numerical results provide the effectiveness of the proposed algorithm.

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In this paper, we propose a spectral projected gradient method for the convex optimization problem with singular solution. By solving the equivalent equation of the gradient function, this method combines the perturbed spectral gradient direction with the projection direction to generate the next iteration point. Under some mild conditions, we establish the global convergence and the local R-linear convergence rate under the local error bound condition. Preliminary numerical tests are given to show that the proposed method works well.

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