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We propose an extended primal-dual algorithm framework for solving a generalnonconvex optimization model. This work is motivated by image reconstructionproblems in a class of nonlinear imaging, where the forward operator can beformulated as a nonlinear convex function with respect to the reconstructedimage. Using the proposed framework, we put forward six specific iterativeschemes, and present their detailed mathematical explanation. We also establishthe relationship to existing algorithms. Moreover, under proper assumptions, weanalyze the convergence of the schemes for the general model when the optimaldual variable regarding the nonlinear operator is non-vanishing. As arepresentative, the image reconstruction for spectral computed tomography isused to demonstrate the effectiveness of the proposed algorithm framework. Byspecial properties of the concrete problem, we further prove the convergence ofthese customized schemes when the optimal dual variable regarding the nonlinearoperator is vanishing. Finally, the numerical experiments show that theproposed algorithm has good performance on image reconstruction for variousdata with non-standard scanning configuration.

An Extended Primal-Dual Algorithm Framework for Nonconvex Problems with Application to Nonlinear Imaging