A Spectral Gradient Projection Method for the Positive Semi-definite Procrustes Problem

This paper addresses the positive semi-deffnite procrustes problem (PSDP). The PSDP corresponds to a least squares problem over the set of symmetric and semi-deffnite positive matrices. These kinds of problems appear in many applications such as structure analysis, signal processing, among others. A non-monotone spectral projected gradient algorithm is proposed to obtain a numerical solution for the PSDP. The proposed algorithm employs the Zhang and Hager's non-monotone technique in combination with the Barzilai and Borwein's step size to accelerate convergence. Some theoretical results are presented. Finally, numerical experiments are performed to demonstrate the effectiveness and efficiency of the proposed method, and comparisons are made with other state-of-the-art algorithms.

Sparse Signal Inversion with Impulsive Noise by Dual Spectral Projected Gradient Method

We consider sparse signal inversion with impulsive noise. There are three major ingredients. The first is regularizing properties; we discuss convergence rate of regularized solutions. The second is devoted to the numerical solutions. It is challenging due to the fact that both fidelity and regularization term lack differentiability. Moreover, for ill-conditioned problems, sparsity regularization is often unstable. We propose a novel dual spectral projected gradient (DSPG) method which combines the dual problem of multiparameter regularization with spectral projection gradient method to solve the nonsmooth l1+l1 optimization functional. We show that one can overcome the nondifferentiability and instability by adding a smooth l2 regularization term to the original optimization functional. The advantage of the proposed functional is that its convex duality reduced to a constraint smooth functional. Moreover, it is stable even for ill-conditioned problems. Spectral projected gradient algorithm is used to compute the minimizers and we prove the convergence. The third is numerical simulation. Some experiments are performed, using compressed sensing and image inpainting, to demonstrate the efficiency of the proposed approach.