Value function of variational regularization ($\Psi (\alpha )={\inf }_u[D(u)+\alpha J(u)]$)

For a nonnegative regulariser $J$, the optimal objective value is non-decreasing and concave in the positive regularization parameter. If $J(0)=0$ and $D(0)<\infty$, comparison with zero makes it finite and a locally Lipschitz function away from parameter zero.

For $0<\alpha <\beta$, choosing global minimizers of $D+\alpha J$ and $D+\beta J$ gives $J(u_{\beta })\le J(u_{\alpha })$ and $D(u_{\beta })\ge D(u_{\alpha })$. The decrease in penalty and increase in data misfit satisfy $\alpha \Delta J\le \Delta D\le \beta \Delta J$.