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.

本文研究了希尔伯特空间中的线性约束最小二乘优化问题，对于该问题，KKT系统不一定可用于分析和计算解。主要目标是为圆锥形正则化方法中的正则化误差开发新的定性和定量稳定性估计。为了实现此目标，我们将稳定性的概念与根据域空间上的正则轨迹和约束空间上的正则状态轨迹定义的一些标量和向量优化问题的可解性关联起来。我们分析了三种优化公式。第一个公式使正则轨迹上的标量目标函数最小化。第二种公式包括矢量，用于优化特定Bishop–Phelps锥的畴空间上的正则轨迹。第三个公式导致矢量优化约束锥的正则化状态轨迹。我们提供了一些数字示例来说明所开发框架的功效。