Given the following dual optimization problem:
$$\min_x \|Ax - y\|_2\quad\text{such that}\quad \|x\|_2 \leq r.$$
What is the minimizer?
Given the Moore-Penrose pseudoinverse $A^+,$ it is evident to me that if $\|A^+ y\| \leq r$, then $A^+ y$ is the minimizer to that problem. But what if $\|A^+ y\| > r?$ Is calculating $r {A^+ y \over \|A^+ y\|}$ enough?
This problem occurs in trust-region algorithms for optimization and is known as the "trust-region subproblem" (sometimes "trust region subproblem.") There are many papers written about efficient and accurate numerical methods for its solution. There's no simple closed form solution.