Constrained-penalized equivalence for total variation denoising

For finite convex total variation and a positive squared-error budget, the Slater condition supplies a multiplier $\lambda \ge 0$ such that every constrained minimizer also minimizes $\mathrm{TV}(u)+\lambda \Vert u-g{\Vert }^2$. Conversely, a minimizer of that penalized objective with $\lambda >0$ minimizes total variation under the budget equal to its own squared error. Complementary slackness proves the first implication; a comparison of objective values proves the second. This does not assert uniqueness of the multiplier or equality of all minimizer sets when $\lambda =0$.