Motivated principally by the low-rank matrix completion problem, we present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. This is accomplished by a new approach to generating ``in-face directions at each iteration, as well as through new choice rules for selecting between in-face and ``regular Frank-Wolfe steps. Our framework for generating in-face directions generalizes the notion of away-steps introduced by Wolfe. In particular, the in-face directions always keep the next iterate within the minimal face containing the current iterate. We present computational guarantees for the new method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply the new method to the matrix completion problem, where low-dimensional faces correspond to low-rank matrices. We present computational results that demonstrate the effectiveness of our methodological approach at producing nearly-optimal solutions of very low rank. On both artificial and real datasets, we demonstrate significant speed-ups in computing very low-rank nearly-optimal solutions as compared to either the Frank-Wolfe method or its traditional away-step variant.
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