Abstract Optimization problems involving the sum of three convex functions have received much attention in recent years, where one is differentiable with Lipschitz continuous gradient, one is composed of a linear operator and the other is proximity friendly. The primal-dual fixed point algorithm is a simple and effective algorithm for such problems. To exploit the second-order derivatives information of the objective function, we propose a primal-dual fixed point algorithm with an adapted metric method. The proposed algorithm is derived from the idea of establishing a generally fixed point formulation for the solution of the considered problem. Under mild conditions on the iterative parameters, we prove the convergence of the proposed algorithm. Further, we establish the ergodic convergence rate in the sense of primal-dual gap and also derive the linear convergence rate with additional conditions. Numerical experiments on image deblurring problems show that the proposed algorithm outperforms other state-of-the-art primal-dual algorithms in terms of the number of iterations.

中文翻译：

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基于自适应度量法求解凸极小化问题的原对偶不动点算法与应用

摘要 近年来，涉及三个凸函数和的优化问题备受关注，其中一个是用Lipschitz连续梯度可微的，一个是由线性算子组成的，另一个是邻近友好的。原始对偶不动点算法是解决此类问题的一种简单有效的算法。为了利用目标函数的二阶导数信息，我们提出了一种具有自适应度量方法的原始对偶定点算法。所提出的算法源自为所考虑问题的解决方案建立一般不动点公式的思想。在迭代参数的温和条件下，我们证明了所提出算法的收敛性。更多，我们建立了原始对偶间隙意义上的遍历收敛率，并在附加条件下推导出线性收敛率。图像去模糊问题的数值实验表明，所提出的算法在迭代次数方面优于其他最先进的原始对偶算法。