Chambolle-Pock's Primal-Dual Method with Mismatched Adjoint

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 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.