Convergence with dynamically selected regularization parameters

Extend the limited-memory MM-GKS method and its convergence theory to prove convergence to the minimum of the smoothed functional while dynamically determining an optimal regularization parameter at each iteration.

Background

The convergence analysis developed in the paper assumes that the regularization parameter is fixed and known. In practical applications, however, the parameter is selected dynamically, for example by generalized cross-validation, within each projected problem. The proposed limited-memory MM-GKS method establishes convergence under a fixed regularization parameter, but the paper does not prove that convergence is preserved when the parameter is updated during the iterations.

Resolving this problem would extend the theoretical guarantees of limited-memory MM-GKS to the parameter-selection regime used in the numerical experiments, thereby providing a convergence result for the complete practical algorithm rather than only for its fixed-parameter specialization. The conclusion identifies this as a central direction for future work.