Nomonotone Spectral Gradient Method for Sparse Recovery

In the paper, we present an algorithm framework for the more general problem of minimizing the sum f(x) + ψ(x), where f is smooth and ψ is convex, but possible nonsmooth. At each step, the search direction of the algorithm is obtained by solving an optimization problem involving a quadratic term with diagonal Hessian and Barzilai-Borwein steplength plus ψ(x). The method with the nomonotone line search techniques is showed to be globally convergent. In particular, if f is convex, we show that the method shares a sublinear global rate of convergence. Moreover, if f is strongly convex, we prove that the method converges R-linearly. Numerical experiments with compressive sense problems show that our approach is competitive with several known methods for some standard `2 − `1 problems.