A modified spectral gradient projection-based algorithm for large-scale constrained nonlinear equations with applications in compressive sensing

To solve large-scale nonlinear equations with convex constraints efficiently and overcome the shortcomings of other algorithms, such as large storage and high complexity, we develop a descent and derivative-free algorithm based on a classical spectral gradient method, the projection technique and a new monotone line search method. The new algorithm has the following features: (1) There is lower storage and derivative-free information; (2) It is suitable to solve large-scale nonlinear equations; (3) Its direction possesses sufficient descent and trust region properties; (4) The spectral parameter is bounded. The global convergence of the new algorithm is proven under some appropriate assumptions. If the local error bound condition is satisfied, the proposed algorithm is linearly convergent. Preliminary numerical results demonstrate that the new algorithm is effective and robust. Furthermore, we also extend this algorithm to solve the sparse signal reconstruction problem and the blurred image recovery problem in compressed sensing.