M-PSGP: a momentum-based proximal scaled gradient projection algorithm for nonsmooth optimization with application to image deblurring

In this study, we focus on investigating a nonsmooth convex optimization problem involving the <i>l</i> ₁-norm under a non-negative constraint, with the goal of developing an inverse-problem solver for image deblurring. Research focused on solving this problem has garnered extensive attention and has had a significant impact on the field of image processing. However, existing optimization algorithms often suffer from overfitting and slow convergence, particularly when working with ill-conditioned data or noise. To address these challenges, we propose a momentum-based proximal scaled gradient projection (M-PSGP) algorithm. The M-PSGP algorithm, which is based on the proximal operator and scaled gradient projection (SGP) algorithm, integrates an improved Barzilai-Borwein-like step-size selection rule and a unified momentum acceleration framework to achieve a balance between performance optimization and convergence rate. Numerical experiments demonstrate the superiority of the M-PSGP algorithm over several seminal algorithms in image deblurring tasks, highlighting the significance of our improved step-size strategy and momentum-acceleration framework in enhancing convergence properties.