The Barzilai-Borwein (BB) method is a popular and efficient algorithm for solving convex optimization problems, which is a gradient method while the step size is selected under some quasi-Newton idea, measured in the Euclidean distance. In this paper, we apply a more general distance measure (e.g., the Bregman divergence) to the BB method. We derive several new BB step sizes formulae and apply them to the mirror descent method and the Frank-Wolfe method. Compared with the two algorithms with traditional BB step sizes, the preliminary numerical experiments for the real data demonstrate that the algorithms with the proposed step sizes are efficient, and can achieve better performance in terms of taking less CPU time to achieve better objective value.
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