Accelerated methods with fastly vanishing subgradients for structured non-smooth minimization

In a real Hilbert space, we study a new class of forward-backward algorithms for structured non-smooth minimization problems. As a special case of the parameters, we recover the method AFB (Accelerated Forward-Backward) that was recently discussed as an enhanced variant of FISTA (Fast Iterative Soft Thresholding Algorithm). Our algorithms enjoy the well-known properties of AFB. Namely, they generate convergent sequences (xn) that minimize the function values at the rate o(n− 2). Another important specificity of our processes is that they can be regarded as discrete models suggested by first-order formulations of Newton-like dynamical systems. This permit us to extend to the non-smooth setting, a property of fast convergence to zero of the gradients, established so far for discrete Newton-like dynamics with smooth potentials only. In specific, as a new result, we show that the latter property also applies to AFB. To prove this stability phenomenon, we develop a technical analysis that can be also useful regarding many other related developments. Numerical experiments are furthermore performed so as to illustrate the properties of the considered algorithms comparing with other existing ones.