= Constrained-penalized equivalence for total variation denoising
For finite convex <total variation> and a positive squared-error budget, the <Slater condition> supplies a multiplier $\lambda\geq0$ such that every constrained minimizer also minimizes $\operatorname{TV}(u)+\lambda\|u-g\|^2$. 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 $\lambda=0$.
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