An efficient conjugate gradient-based algorithm for unconstrained optimization and its projection extension to large-scale constrained nonlinear equations with applications in signal recovery and image denoising problems

Abstract

In this paper, a hybrid version of the Rivaie–Mustafa–Ismail–Leong, RMIL for short, conjugate gradient method is proposed for unconstrained optimization problem. The search direction fulfills the sufficient descent condition and trust region property at each iteration, independent of any line search. We establish the global convergence of the proposed algorithm under normal assumptions and Wolfe line search. To move forward, incorporating with derivative-free projection technique, the proposed conjugate coefficient is extended to deal with nonlinear monotone equations. Without Lipschitz continuity assumption, the global convergence is proved with mild conditions. The numerical results for both unconstrained optimization instances and large-scale nonlinear equations prove the efficiency of the proposed methods with less computation cost. In particular, we also explore the practical applications on signal recovery and image denoising problems.