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The primal-dual method of Chambolle and Pock is a widely used algorithm tosolve various optimization problems written as convex-concave saddle pointproblems. Each update step involves the application of both the forward linearoperator and its adjoint. However, in practical applications like computerizedtomography, it is often computationally favourable to replace the adjointoperator by a computationally more efficient approximation. This leads to anadjoint mismatch in the algorithm.  In this paper, we analyze the convergence of Chambolle-Pock's primal-dualmethod under the presence of a mismatched adjoint in the strongly convexsetting. We present an upper bound on the error of the primal solution andderive stepsizes and mild conditions under which convergence to a fixed pointis still guaranteed. Furthermore we show linear convergence similar to theresult of Chambolle-Pock's primal-dual method without the adjoint mismatch.Moreover, we illustrate our results both for an academic and a real-worldinspired application.

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