```markdown
### Image Description

The image is a slide titled "Duality: SOCP" from a presentation on convex optimization. It describes the process of introducing new variables to rewrite an optimization problem. The problem is structured as follows:

- **Objective:** Minimize \( c^T x \)
- **Subject to:**
  - \( \|y_i\|_2 \leq t_i \), for \( i = 1, \ldots, m \)
  - \( y_i = A_i x + b_i \)
  - \( t_i = c_i^T x + d_i \), for \( i = 1, \ldots, m \)

The slide also presents the Lagrangian of the problem:

\[ 
L(x, y, t, \lambda, \nu, \mu) = c^T x + \sum_{i=1}^{m} \lambda_i (\|y_i\|_2 - t_i) + \sum_{i=1}^{m} \nu_i^T (y_i - A_i x - b_i) + \sum_{i=1}^{m} \mu_i (t_i - c_i^T x - d_i) 
\]

This expands to:

\[ 
= \left(c - \sum_{i=1}^{m} A_i^T \nu_i - \sum_{i=1}^{m} \mu_i c_i\right)^T x + \sum_{i=1}^{m} (\lambda_i \|y_i\|_2 + \nu_i^T y_i) + \sum_{i=1}^{m} (-\lambda_i + \mu_i) t_i 
- \sum_{i=1}^{n} (b_i^T \nu_i + d_i \mu_i)
\]

The slide is numbered as 35 out of 49 and is authored by A. d’Aspremont, part of a course on Convex Optimization.
```