Large scale nonsmooth convex optimization is a common problem for a range of computational areas including machine learning and computer vision. Problems in these areas contain special domain structures and characteristics. Special treatment of such problem domains, exploiting their structures, can significantly improve the the computational burden. We present a weighted Mirror Descent method to solve optimization problems over a Cartesian product of convex sets. The algorithm uses a nonlinear weighted projection scheme to generate feasible search directions. This projection scheme uses weighting parameters that, eventually, lead to the optimal step-size strategy for every projection on a corresponding subset. We demonstrate the efficiency of the algorithm by solving the Markov Random Fields optimization problem. In particular, we employ the log-entropy distance and the Euclidean distance in the proposed algorithm. Promising experimental results demonstrate the effectiveness of the proposed method.
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