An Inertial-Relaxed Three-Term Conjugate Gradient Projection Method for Large-Scale Unconstrained Nonlinear Pseudo-Monotone Equations with Applications

In this paper, a three-term conjugate gradient projection method that employs both inertial and relaxed techniques is proposed to find approximate solutions for unconstrained nonlinear pseudo-monotone equations. The search direction generated at each iteration possesses the sufficient descent and trust region properties independent of the line search technique used. The global convergence of the proposed method is shown without the Lipschitz continuity of the underlying mapping. Moreover, the asymptotic and non-asymptotic convergence rates in terms of iteration complexity are established under the local Lipschitz continuity assumption. To our knowledge, this is the first time in the literature that an iteration-complexity analysis has been conducted for inertial-relaxed gradient-type projection methods. Numerical experiments on large-scale benchmark test problems are conducted to demonstrate the effectiveness and efficiency of the proposed algorithm. Furthermore, the applicability and practicality of the proposed method are also verified by applying it to solve sparse signal restoration problems.