A new smoothing modified three-term conjugate gradient method for l 1 $l_{1}$ -norm minimization problem

Abstract

We consider a kind of nonsmooth optimization problems with -norm minimization, which has many applications in compressed sensing, signal reconstruction, and the related engineering problems. Using smoothing approximate techniques, this kind of nonsmooth optimization problem can be transformed into a general unconstrained optimization problem, which can be solved by the proposed smoothing modified three-term conjugate gradient method. The smoothing modified three-term conjugate gradient method is based on Polak–Ribière–Polyak conjugate gradient method. For the Polak–Ribière–Polyak conjugate gradient method has good numerical properties, the proposed method possesses the sufficient descent property without any line searches, and it is also proved to be globally convergent. Finally, the numerical experiments show the efficiency of the proposed method.

A New Non-Convex Approach for Compressive Sensing Mri

“…It has been shown in [29] that under certain RIP condition of A, L q -norm minimization algorithms require fewer sampling data but gain a better recovery performance than L 1 -norm minimization algorithms. Moreover, the sufficient conditions in terms of RIP for L qnorm minimization are weaker than those for L 1 -norm minimization [30,31]. However, in general, relative to L 1 -norm minimization, L q -norm minimization is more difficult to directly tackle due to its nonsmoothing.…”