A non-monotonic method for large-scale non-negative least squares

We present a new algorithm for nonnegative least-squares (NNLS). Our algorithm extends the unconstrained quadratic optimization algorithm of Barzilai and Borwein (BB) (J. Barzilai and J. M. Borwein; Two-Point Step Size Gradient Methods. IMA J. Numerical Analysis; 1988.) to handle nonnegativity constraints. Our extension differs in several basic aspects from other constrained BB variants. The most notable difference is our modified computation of the BB stepsize that takes into account the nonnegativity constraints. We further refine the stepsize computation by introducing a stepsize scaling strategy that, in combination with orthogonal projections onto the nonnegative quadrant, yields an efficient NNLS algorithm. We compare our algorithm with both established convex solvers and a specialized NNLS method: on several synthetic and real-world datasets we observe highly competitive empirical performance.