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. We present an upper bound on the error of the primal solution and derive step-sizes and mild conditions under which convergence to a fixed point is still guaranteed. Furthermore we present convergence rates similar to these of Chambolle-Pock's primal-dual method without the adjoint mismatch. Moreover, we illustrate our results both for an academic and a real-world application.
翻译:Chambolle和Pock的原始双向方法是一种被广泛使用的算法,用于解决各种优化问题,这些算法被写成共聚点问题。每个更新步骤都涉及前线操作员及其辅助操作员的应用。然而,在计算机化断层法等实际应用中,通常可以计算出一种有利的方法,用一种效率更高的计算近似法取代联合操作员。这导致算法中的一种不匹配。在本文中,我们分析了Chambolle-Pock的原始双向方法在存在不匹配的对接点的情况下的趋同情况。我们展示了原始解决方案错误的上限,并得出了与固定点趋同仍得到保证的继体大小和温度条件。此外,我们展示了与Campolle-Pock的原始-双向方法相似的趋同率,而没有连接不匹配的不匹配。此外,我们用一个学术和真实世界应用程序来说明我们的结果。