  This paper is devoted to minimizing the sum of a smooth function and a
nonsmooth $\ell_1$-regularized term. This problem as a special cases includes
the $\ell_1$-regularized convex minimization problem in signal processing,
compressive sensing, machine learning, data mining, etc. However, the
non-differentiability of the $\ell_1$-norm causes more challenging especially
in large problems encountered in many practical applications. This paper
proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each
iteration, the generated search direction enjoys descent property and can be
easily derived by minimizing a local approximal quadratic model and
simultaneously taking the favorable structure of the $\ell_1$-norm. Moreover, a
nonmonotone line search technique is incorporated to find a suitable stepsize
along this direction. The algorithm is easily performed, where the values of
the objective function and the gradient of the smooth term are required at
per-iteration. Under some conditions, the proposed algorithm is shown to be
globally convergent. The limited experiments by using some nonconvex
unconstrained problems from CUTEr library with additive $\ell_1$-regularization
illustrate that the proposed algorithm performs quite well. Extensive
experiments for $\ell_1$-regularized least squares problems in compressive
sensing verify that our algorithm compares favorably with several
state-of-the-art algorithms which are specifically designed in recent years.
