On Regularized Least Norm Problems

This paper investigates the regularized least norm problem \[ \text{minimize }F(\mathbf{x}) = (\varepsilon /s) \| \mathbf{x} \|_s^s + \| A\mathbf{x} - \mathbf{b} \|_p , \] where $\varepsilon $ is a positive constant, $1 < s < \infty $, and $1 < p < \infty $. Let $\mathbf{x}_\varepsilon $ denote the solution that corresponds to a given value of $\varepsilon $, and let $\mathbf{x}^* $ denote the minimum $l_s $ norm solution of the unregularized least $l_p $ norm problem. It is shown that $\mathbf{x}_\varepsilon $ is a continuous function of $\varepsilon $, $\|\mathbf{x}_\varepsilon \|_s \leqq \| \mathbf{x}^*\|_s$, $\lim_{\varepsilon \to \infty } \mathbf{x}_\varepsilon = \mathbf{0}$, and $\lim_{\varepsilon \to 0} \mathbf{x}_\varepsilon = \mathbf{x}^* $. Furthermore, if the system $A\mathbf{x} = \mathbf{b}$ is solvable then there exists a positive constant $\delta $ such that $\mathbf{x}_\varepsilon = \mathbf{x}^* $ for all $\varepsilon \in ( 0,\delta ]$. The question of whether $\mathbf{x}_\varepsilon = \mat...