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A new algorithm is presented for efficiently solving image reconstruction problems that arise in partially parallel magnetic resonance imaging. This algorithm minimizes an objective function of the form φ(Bu) + 1/2||FpSu - f||2, where φ is the regularization term which may be nonsmooth. In image reconstruction, the φ term corresponds to total variation smoothing and/or L1 regularization term. The least square term 1/2||FpSu - f||2 is the fidelity term. In our application, f represents undersampled data from a partially parallel imaging (PPI) system. The proposed algorithm is a generalization of the Bregman operator splitting algorithm with variable stepsize (BOSVS) in which the previous Barzilai-Borwein (BB) step is replaced by a cyclic BB (CBB) step, and an L1 term Ψ is added to the energy function. Experimental results on clinical partially parallel imaging data are given.