Line search methods are a prominent class of iterative methods to solve unconstrained minimization problems. These methods produce new iterates utilizing a suitable step size after determining proper directions for minimization. In this paper we propose a semi-monotone line search technique based on the Goldstein quotient for dealing with convex non-smooth optimization problems. The method allows to employ large step sizes away from the optimum thus improving the efficacy compared to standard Goldstein approach. For the presented line search method, we prove global convergence to a stationary point and local R-linear convergence rate in strongly convex cases. We report on some experiments in compressed sensing. By comparison with several state-of-the-art algorithms in the field, we demonstrate the competitive performance of the proposed approach and specifically its high efficiency.