```markdown
**Title:** Duality: SOCP

The slide discusses conditions for boundedness in a Second-Order Cone Programming (SOCP) context.

- The minimum over \( x \) is bounded below if and only if:
  \[
  \sum_{i=1}^{m} (A_i^T \nu_i + \mu_i c_i) = c.
  \]

- To minimize over \( y_i \), it is noted that:
  \[
  \inf_{y_i} (\lambda_i \|y_i\|_2 + \nu_i^T y_i) = 
  \begin{cases} 
  0 & \|\nu_i\|_2 \leq \lambda_i \\
  -\infty & \text{otherwise}.
  \end{cases}
  \]

- The minimum over \( t_i \) is bounded below if and only if \( \lambda_i = \mu_i \).

**Source:** A. d'Aspremont. Convex Optimization M2. Slide 36/49.
```
