In this paper, we consider the convergence rate of the alternating direction method of multipliers for solving the nonconvex separable optimization problems. Based on the error bound condition, we prove that the sequence generated by the alternating direction method of multipliers converges locally to a critical point of the nonconvex optimization problem in a linear convergence rate, and the corresponding sequence of the augmented Lagrangian function value converges in a linear convergence rate. We illustrate the analysis by applying the alternating direction method of multipliers to solving the nonconvex quadratic programming problems with simplex constraint, and comparing it with some state-of-the-art algorithms, the proximal gradient algorithm, the proximal gradient algorithm with extrapolation, and the fast iterative shrinkage–thresholding algorithm.

中文翻译：

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非凸可分优化问题乘法器交替方向法的局部线性收敛

本文考虑乘法器交替方向法求解非凸可分优化问题的收敛速度。基于误差界条件，证明乘法器交替方向法生成的序列以线性收敛速度局部收敛于非凸优化问题的临界点，增广拉格朗日函数值的对应序列收敛于线性收敛速度。我们通过应用乘法器的交替方向法解决具有单纯形约束的非凸二次规划问题来说明分析，并将其与一些最先进的算法，近端梯度算法，带外推的近端梯度算法进行比较，