A Cyclic Projected Gradient Method

In recent years, convex optimization methods were successfully applied for various im-age processing tasks and a large number of first-order methods were designed to minimize the corresponding functionals. Interestingly, it was shown recently in [24] that the sim-ple idea of so-called “superstep cycles ” leads to very efficient schemes for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image com-pression tasks. The “superstep cycles ” approach is similar to the nonstationary (cyclic) Richardson method which has been around for over sixty years. In this paper, we investigate the incorporation of superstep cycles into the projected gradient method. We show for two problems in compressive sensing and image processing, namely the LASSO approach and the Rudin-Osher-Fatemi model that the resulting simple cyclic projected gradient algorithm can numerically compare with various state-of-the-art first-order algorithms. However, due to the nonlinear projection within the algorithm convergence proofs even under restrictive assumptions on the linear operators appear to be hard. We demonstrate the difficulties by studying the simplest case of a two-cycle algorithm in R2 with projections onto the Euclidean ball.