MGProx

Reproduction for 2302.04077 which aims to minimize the sum of a $\mu_0$-strongly convex, $L_0$-smooth $f_0$ and a convex, nonsmooth, lower semi-continuous and seperable $g_0$. $$ \begin{equation} x^*=\mathop{\arg\min}{x\in\R^n}\ F_0(x):=f_0(x)+g_0(x)\tag{1} \end{equation} $$ proximal mapping: $$ \begin{equation} x{k+1}=\mathop{\arg\min}_u\ \frac 1 2\Vert u-x_k\Vert_2^2+g_0(u)\tag{2} \end{equation} $$

EOP

Elastic Obstacle Problem(EOP) describes the shape of an elastic membrane covering an obstacle $\phi$. We discretize a 2-dimensional shifted aEOP on truncated sine wave. $$ \begin{equation} \min_x \frac 1 2 \Braket{Q_0x,x}+\Braket{p_0,x}+i_+(x)\label{eq1}\tag{3} \end{equation} $$ proximal mapping: $$ \begin{aligned} x_{k+1}&=\mathop{\arg\min}u\ \frac 1 2\Vert u-x_k\Vert_2^2+i+(u)\ &=\max(x_k,0) \end{aligned}\tag{4} $$

LASSO

problem $$ \min_x\ \frac12\Vert Ax-b\Vert_2^2+\lambda\Vert x\Vert_1\tag{5} $$ proximal mapping $$ \begin{aligned} x_{k+1}&=\mathop{\arg\min}_u\ \frac 1 2\Vert u-x_k\Vert_2^2+\lambda\Vert u\Vert_1\ &=\operatorname{sign}(x_k)\odot\max(|x_k|-\lambda,0) \end{aligned}\tag{6} $$