  We propose an extended primal-dual algorithm framework for solving a general
nonconvex optimization model. This work is motivated by image reconstruction
problems in a class of nonlinear imaging, where the forward operator can be
formulated as a nonlinear convex function with respect to the reconstructed
image. Using the proposed framework, we put forward six specific iterative
schemes, and present their detailed mathematical explanation. We also establish
the relationship to existing algorithms. Moreover, under proper assumptions, we
analyze the convergence of the schemes for the general model when the optimal
dual variable regarding the nonlinear operator is non-vanishing. As a
representative, the image reconstruction for spectral computed tomography is
used to demonstrate the effectiveness of the proposed algorithm framework. By
special properties of the concrete problem, we further prove the convergence of
these customized schemes when the optimal dual variable regarding the nonlinear
operator is vanishing. Finally, the numerical experiments show that the
proposed algorithm has good performance on image reconstruction for various
data with non-standard scanning configuration.
