I have difficulties converting a constraint to a SOC constraint. Here is the full problem:
\begin{align} \text{minimize} &\quad \alpha \nonumber\\ \text{subject to} &\quad \sum_{P_j \in \Pi_i} x_{ij} = 1 & (1)\\ &\quad y_{ik} = \sum_{P_j \in \Pi_i} x_{ij} \mathbf{I}_{L_k \in P_j} & (2)\\ &\quad \sum_{L_k \in P_j} \varepsilon_k \le \epsilon & (3)\\ &\quad \big( 2 \ln\frac{1}{\varepsilon_k} \big) \sum_{F_i} \sigma^2_i y^2_{ik} \le \Big( \alpha C_k - \sum_{F_i} \mu_i y_{ik} \Big)^2 & (4)\\ &\quad 0 \le x_{ij} \le 1 \\ &\quad 0 \le y_{ik} \le 1 \\ &\quad 0 \le \varepsilon_k \le 1 \end{align}
Problem is Constraint (4), which can be stated as below: \begin{equation} \theta \sum_{i} y_i^2 \le u^2, \end{equation} where $0 \le y_i \le 1$, $\theta \ge 0$, and $u \ge 0$ are optimization variables. If $\theta$ was not present, the constraint would be a standard SOC constraint.
I have the following, \begin{align} \theta y_i^2 \le z_i^2 \Rightarrow \sum_{i} z_i^2 \le u^2 \end{align} which is now a SOC constraint, if the following is correct: \begin{align} &\theta y_i^2 \le z_i^2 \\ &\equiv y_i^2 \le \frac{1}{\theta} z_i^2 \\ &\equiv \begin{cases} y_i^2 \le t z_i^2\\ \frac{1}{\theta} \le t \end{cases} \qquad(1)\\ &\equiv \begin{cases} y_i^2 \le t w_i\\ z_i^2 \le w_i\\ 1 \le \theta t \end{cases} \qquad(2) \end{align}
The final system of inequalities can be easily converted to standard cone constraints \begin{equation} \equiv \begin{cases} 4 y_i^2 + (t - w_i)^2 \le (t + w_i)^2\\ 4 z_i^2 + (1 - w_i)^2 \le (1+w_i)^2\\ 4 + (\theta - t)^2 \le (\theta + t)^2 \end{cases} \end{equation}
Question: Are the inequalities in steps (1) and (2) equivalent to the original constraint? Any comment is appreciated.