In this paper, we study the iteration complexity of a block coordinate gradient descent (BCGD) method with a cyclic rule for solving convex optimization problems. We propose a new Lipschitz continuity-like assumption and show that the iteration complexity for the proposed BCGD method can be improved to O( max{M, L} ε ), where M is the constant in the proposed assumption, L is the usual Lipschitz constant for the gradient of the objective function, and ε > 0 is the required precision. In addition, we analyze the relation between M and L, and prove that, in the worst case, M ≤ √ NL, where N is the number of blocks.

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