Control proximal gradient algorithm for image $$\ell _1$$ ℓ 1 regularization

Abstract

We consider a control proximal gradient algorithm (CPGA) for solving the minimization of a nonsmooth convex function. In particular, the convex function is an $$\ell _1$$ regularized least squares function derived by the discretized $$\ell _1$$ norm models arising in image processing. This proximal gradient algorithm with the control step size is attractive due to its simplicity. However, this method is also known to converge quite slowly. In this paper, we present a fast control proximal gradient algorithm by adding Nesterov step. It preserves the computational simplicity of proximal gradient methods with a convergence rate $$1/k^2$$ . It is proven to be better than the existing methods both theoretically and practically. The initial numerical results for performing the image deblurring demonstrate the efficiency of CPGA when it is compared to the existing fast iterative shrinkage-thresholding algorithms.