A new nonmonotone spectral residual method for nonsmooth nonlinear equations

In this paper, a new spectral residual method is proposed to solve systems of large-scale nonlinear equations, where the steplength is obtained by minimizing the residue of an approximate secant equation. Especially, the new steplength can be directly applied into solving strictly convex quadratic function. Combined with a new nonmonotone line search strategy, a new derivative-free algorithm, called a nonmonotone spectral residual algorithm (NSRA), is developed. Under mild assumptions, global convergence is established for locally Lipschitz continuous nonlinear systems. Compared with the state-of-the-art algorithms available in the literatures, the new algorithm is more efficient in solving large-scale benchmark test problems.