An Accelerated Spectral CG‐Type Projection Method With Restart Strategy for Nonlinear Monotone Equations and Applications in Compressed Sensing

ABSTRACT In this study, we present an accelerated spectral conjugate gradient‐type projection method to solve the system of nonlinear monotone equations. Compared to traditional methods for solving such equations, this method utilizes three‐step successive information to produce inertial iterates. The proposed composite search direction framework incorporates an effective restart strategy, eliminating the need for additional conditions, including the specification of conjugate parameter, and satisfies conditions for the sufficient descent condition and trust region property. Establishing the theoretical convergence of the present method does not require Lipschitz continuity. Furthermore, we analyze the asymptotic and non‐asymptotic convergence rates in terms of iteration complexity. To evaluate effectiveness, comparative tests are conducted against existing methods using problems related to the system of nonlinear equations. Moreover, the practicality of the presented method is demonstrated through applications in compressed sensing.