= Second-order cone reformulation of one-sided quadratic denoising
Let $u,g\in\mathbb R^n$, $G\in\mathbb R^{m\times n}$ and $\lambda\ge0$. The objective
$$
\|u-g\|_2+\lambda\sum_{i=1}^m((Gu)_i)_+^2
$$
has the exact <second-order cone program> lift
$$
\min\ t+\lambda\sum_iq_i,
\qquad (u-g,t)\in\mathcal L_{n+1},\qquad
\left(\frac{q_i-1}{2},a_i,\frac{q_i+1}{2}\right)\in\mathcal L_3,
\qquad a_i\ge0,\quad a_i\ge(Gu)_i.
$$
The cone ordering here is $(v,s)$ with $\|v\|_2\le s$, matching <second-order cone>. The small cone is equivalent to $q_i\ge a_i^2$. Hence every lifted feasible objective bounds the original one. Choosing $a_i=((Gu)_i)_+$, $q_i=a_i^2$ and $t=\|u-g\|_2$ proves equality. The sign $\lambda\ge0$ is necessary for this bounding argument; a negative penalty coefficient would allow increasing $q_i$ to lower the lifted objective without bound.
Using a parameter-two barrier for each Lorentz block and a parameter-one logarithmic barrier for each scalar nonnegative slack gives total barrier parameter $2+2m+2m=4m+2$. The number of derivative rows is $m$, which need not equal the signal dimension $n$. The fidelity norm remains unsquared and negative derivative values remain unpenalized.
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