A modified Liu-Storey scheme for nonlinear systems with an application to image recovery

Like the Polak-Ribi`ere-Polyak (PRP) and Hestenes-Stiefel (HS) meth-ods, the classical Liu-Storey (LS) conjugate gradient scheme is widely be-lieved to perform well numerically. This is attributed to the in-built capa-bility of the method to conduct a restart when a bad direction is encoun-tered. However, the scheme’s inability to generate descent search direc-tions, which is vital for global convergence, represents its major shortfall. In this article, we present an LS-type scheme for solving system of mono-tone nonlinear equations with convex constraints. The scheme is based on the approach by Wang et al. (2020) and the projection scheme by Solodov and Svaiter (1998). The new scheme satisfies the important condition for global convergence and is suitable for non-smooth nonlinear problems. Fur-thermore, we demonstrate the method’s application in restoring blurry im-ages in compressed sensing. The scheme’s global convergence is established under mild assumptions and preliminary numerical results show that the proposed method is promising and performs better than two recent meth-ods in the literature.