论文标题
在某些原始偶算算法的改善条件下
On the improved conditions for some primal-dual algorithms
论文作者
论文摘要
$ f（\ mathbf {x}）+g（\ mathbf {x}）+h（\ MathBf {A} \ Mathbf {x}）$上$ f（\ mathbf {x}）$ y $ f（\ mathbf {x}）$ a $ f（\ mathbf {x}）$上的概念最小化的凸率最小化。 \ Mathbb {r}^m $，在文献中已经进行了充分研究。通过考虑问题的原始二重式最优性，从不同角度（例如单调操作员方案和固定点理论）提出了许多算法。在本文中，我们从一种基础算法开始，以揭示几种算法（例如AFBA，PD3O和Chambolle-Pock）之间的联系。然后，我们证明其在与线性操作员相关的放松假设下的收敛性，并表征了对原始和二步骤的一般约束。结果改善了AFBA步骤的上限，并表明Chambolle-Pock是$ f = 0 $的基本算法的特殊情况，可以将双重迭代的步骤提高到先前证明的$ 4/3 $。
The convex minimization of $f(\mathbf{x})+g(\mathbf{x})+h(\mathbf{A}\mathbf{x})$ over $\mathbb{R}^n$ with differentiable $f$ and linear operator $\mathbf{A}: \mathbb{R}^n\rightarrow \mathbb{R}^m$, has been well-studied in the literature. By considering the primal-dual optimality of the problem, many algorithms are proposed from different perspectives such as monotone operator scheme and fixed point theory. In this paper, we start with a base algorithm to reveal the connection between several algorithms such as AFBA, PD3O and Chambolle-Pock. Then, we prove its convergence under a relaxed assumption associated with the linear operator and characterize the general constraint on primal and dual stepsizes. The result improves the upper bound of stepsizes of AFBA and indicates that Chambolle-Pock, as the special case of the base algorithm when $f=0$, can take the stepsize of the dual iteration up to $4/3$ of the previously proven one.
