Given the vectors $h,\theta_{i,j}\in \mathbb R^{m}$, the matrix $\mathcal P\in \mathbb R^{m\times m}$, the symmetric matrices $U\in \mathbb R^{m\times m},S\in \mathbb R^{(m+1)\times(m+1)}$, scalars $\omega,\rho \in \mathbb R_+$, $a_{i,j}, b_{i,j}\in \mathbb R$, the optimization problem w.r.t. the decision variable $u\in \mathbb R^m$ is $$ \min_u \; {h^\top}u + \|\mathcal P u\|_1 + \delta_{\mathcal U}(u) + {\frac{\omega}{2}} {\left\| {S} - \begin{pmatrix} U & u \\ u^\top & 1 \end{pmatrix} \right\|}^2 + \frac{\rho}{2} \sum\limits_{i=1}^{M} \sum\limits_{j=1}^N \left\{ \left( {\theta}_{i,j}^\top u + {a}_{i,j} \right)^2 - b_{i,j}^2 \right\}. $$
I wonder how to solve this optimization problem. Note that $\delta_{\mathcal U}(u)$ is an indicator function that $\delta_{\mathcal U}(u) = \begin{cases}0 & \text{if } u\in \mathcal U, \\ \infty & \text{otherwise}\end{cases}$. In fact, the set $\mathcal U = [\underline u, \overline u]$ defines the upper and lower bound of the variable $u$.