An IDFPM-based algorithm without Lipschitz continuity to constrained nonlinear equations for sparse signal and blurred image restoration problems

Abstract

The derivative-free projection method (DFPM) is widely used to solve constrained nonlinear equations. To guarantee the convergence of the derivative-free projection method, the mapping should be Lipschitz continuous, which is a strict requirement in theory. Hence, it is interesting to design the new DFPM that possesses nice convergence under weaker theoretical hypothesis. In this paper, we propose an inertial DFPM-based algorithm (named IDFPM), in which the inertial extrapolation step is embedded in the design for the search direction. The global convergence of the proposed algorithm is obtained without the Lipschitz continuity of the mapping. Numerical experiments are carried out for two kinds of problems. The one consists of eight test problems from classical literature and the compressed sensing model. The numerical results show that the proposed algorithm is promising.