Stability analysis of conically perturbed linearly constrained least-squares problems by optimizing the regularized trajectories

Abstract

This paper studies linearly constrained least-square optimization problems in Hilbert spaces for which the KKT system is not necessarily available to analyze and compute the solution. The primary objective is to develop new qualitative and quantitative stability estimates for the regularization error in the conical regularization approach. To attain this goal, we associate the notion of stability with the solvability of some scalar and vector optimization problems defined in terms of the regularized trajectory on the domain space and the regularized state trajectory on the constraint space. We analyze three optimization formulations. The first formulation minimizes a scalar objective function over the regularized trajectory. The second formulation consists of vector optimizing the regularized trajectory on the domain space for a specific Bishop–Phelps cone. The third formulation results in vector optimizing the regularized state trajectory for the constraint cone. We provide numerical examples to illustrate the efficacy of the developed framework.