```markdown
## Image Description

**Title:** Equality constrained analytic centering

The image presents a slide on equality constrained optimization problems, specifically focusing on analytic centering.

### Content:

- **Primal Problem:** Minimize \(-\sum_{i=1}^{n} \log x_i\) subject to \(Ax = b\).
- **Dual Problem:** Maximize \(-b^T \nu + \sum_{i=1}^{n} \log (A^T \nu)_i + n\).

### Additional Information:

- The slide discusses three methods for an example where \(A \in \mathbb{R}^{100 \times 500}\), with different starting points.
- **Method 1:** Newton method with equality constraints (requires \(x^{(0)} \succ 0\), \(Ax^{(0)} = b\)).

### Graph:

- **Y-axis:** \(f(x^{(k)}) - p^*\) ranging from \(10^{-10}\) to \(10^5\).
- **X-axis:** Iteration number \(k\) from 0 to 20.
- The graph shows convergence behavior of the Newton method with equality constraints.

### Footer:

- **Author:** A. d'Aspremont
- **Course:** Convex Optimization M2
- **Slide Number:** 46/53
```
