Nomonotone spectral gradient method for sparse recovery

In the paper, we present an algorithm framework for the more general problem of minimizing the sum $f(x)+\psi(x)$, where $f$ is smooth and $\psi$ 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 $ \psi(x)$. The nonmonotone strategy is combined with -Borwein steplength to accelerate the convergence process. 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 $l_2-l_1$ problems.