In this paper, a new procedure, called generalized shrinkage conjugate gradient (GSCG), is presented to solve the 1-regularized convex minimization problem. In GSCG, we present a new descent condition. If such a condition holds, an efficient descent direction is presented by an attractive combination of a generalized form of the conjugate gradient direction and the ISTA descent direction. Otherwise, ISTA is improved by a new step-size of the shrinkage operator. The global convergence of GSCG is established under some assumptions and its sublinear (R-linear) convergence rate in the convex (strongly convex) case. In numerical results, the suitability of GSCG is evaluated for compressed sensing and image debluring problems on the set of randomly generated test problems with dimensions n ∈ {210, . . . , 217} and some images, respectively, in Matlab. These numerical results show that GSCG is efficient and robust for these problems in terms of the speed and ability of the sparse reconstruction in comparison with several state-of-the-art algorithms.
