论文标题
Chambolle-Pock的原始偶对偶的方法与不匹配的伴随
Chambolle-Pock's Primal-Dual Method with Mismatched Adjoint
论文作者
论文摘要
Chambolle和Pock的原始偶二方法是一种广泛使用的算法，用于解决作为凸形 - concave鞍点问题所写的各种优化问题。每个更新步骤都涉及向前线性操作员及其伴随的应用。但是，在诸如计算机断层扫描之类的实际应用中，通常在计算上可以通过计算更有效的近似替换伴随运算符。这导致算法中的伴随不匹配。 在本文中，我们分析了Chambolle-Pock的原始偶对偶的收敛性，在强烈凸设置中存在不匹配的伴随的情况下。我们在原始解决方案的误差上提出了上限，并得出了得出的步骤和轻度条件，在该误差下仍然可以保证收敛到固定点。此外，我们显示的线性收敛类似于Chambolle-Pock的原始偶对偶的结果，而没有伴随不匹配。此外，我们为学术和实际启发的应用程序说明了我们的结果。
The primal-dual method of Chambolle and Pock is a widely used algorithm to solve various optimization problems written as convex-concave saddle point problems. Each update step involves the application of both the forward linear operator and its adjoint. However, in practical applications like computerized tomography, it is often computationally favourable to replace the adjoint operator by a computationally more efficient approximation. This leads to an adjoint mismatch in the algorithm. In this paper, we analyze the convergence of Chambolle-Pock's primal-dual method under the presence of a mismatched adjoint in the strongly convex setting. We present an upper bound on the error of the primal solution and derive stepsizes and mild conditions under which convergence to a fixed point is still guaranteed. Furthermore we show linear convergence similar to the result of Chambolle-Pock's primal-dual method without the adjoint mismatch. Moreover, we illustrate our results both for an academic and a real-world inspired application.
