In this paper, we present an efficient method for nonnegative matrix factorization based on the alternating nonnegative least squares framework. Our approach adopts a monotone projected Barzilai–Borwein (MPBB) method as an essential subroutine where the step length is determined without line search.(The Lipschitz constant of the gradient is exploited to accelerate convergence. Global convergence of the proposed MPBB method is established. Numerical results are reported to demonstrate the efficiency of our algorithm)