Iterative regularization of parameter identification problems by sequential quadratic programming methods

The aim of this paper is to design and to analyse sequential quadratic programming (SQP) methods as iterative regularization methods for ill-posed parameter identification problems. We discuss two variants of the original SQP algorithm, in which an additional stabilizer ensures the strict convexity and well posedness of the quadratic programming problems that have to be solved in each step of the iteration procedure. We show that the SQP problems are equivalent to stable saddle-point problems, which can be analysed by standard methods. In addition, the investigation of these saddle-point problems offers new possibilities for the numerical treatment of the identification problem compared to standard numerical methods for inverse problems. One of the resulting iteration algorithms, called the Levenberg-Marquardt SQP method, is analysed with respect to convergence and regularizing properties under an appropriate choice of the stopping index depending on the noise level. Finally, we show that the conditions needed for convergence are fulfilled for several important types of applications and we test the convergence behaviour in numerical examples.