 Residual method for variational regularization

Given $\Vert f_{\delta }-f\Vert \le \delta$, the residual method minimizes a penalty $J(u)$ subject to $\Vert Au-f_{\delta }\Vert \le \delta$. Every exact solution is feasible, so a reconstruction $u_{\delta }$ has penalty no larger than that of any J-minimizing solution.