Abstract

In this paper, we propose a hybrid Bregman alternating direction method of multipliers for solving the linearly constrained difference-of-convex problems whose objective can be written as the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous concave function. At each iteration, we choose either subgradient step or proximal step to evaluate the concave part. Moreover, the extrapolation technique was utilized to compute the nonsmooth convex part. We prove that the sequence generated by the proposed method converges to a critical point of the considered problem under the assumption that the potential function is a Kurdyka–Łojasiewicz function. One notable advantage of the proposed method is that the convergence can be guaranteed without the Lischitz continuity of the gradient function of concave part. Preliminary numerical experiments show the efficiency of the proposed method.

中文翻译：

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线性约束凸差分问题的乘数的混合Bregman交替方向方法

摘要

在本文中，我们提出了一种混合的Bregman交替方向乘子方法，用于解决线性约束的凸差问题，其目标可以写为具有Lipschitz梯度的光滑凸函数，适当的闭合凸函数和连续凹函数。在每次迭代中，我们选择次梯度步骤或近端步骤来评估凹入部分。此外，外推技术被用来计算不光滑的凸部。我们证明，在潜在函数为Kurdyka–Łojasiewicz函数的假设下，所提出的方法生成的序列收敛到所考虑问题的临界点。所提出的方法的一个显着优点是，可以保证收敛，而无需凹部的梯度函数的Lischitz连续性。初步的数值实验表明了该方法的有效性。