```markdown
### Image Description

**Title:** Central path

The slide discusses the concept of the central path in optimization. 

- **Central Path Definition:** For \( t > 0 \), \( x^\star(t) \) is defined as the solution to the optimization problem:
  \[
  \text{minimize } \, t f_0(x) + \phi(x) \\
  \text{subject to } \, Ax = b
  \]
  It is assumed that \( x^\star(t) \) exists and is unique for each \( t > 0 \).

- **Central Path Set:** The central path is represented as \( \{ x^\star(t) \, | \, t > 0 \} \).

- **Example:** An example of the central path for a linear program (LP) is provided:
  \[
  \text{minimize } \, c^T x \\
  \text{subject to } \, a_i^T x \leq b_i, \, i = 1, \ldots, 6
  \]
  The hyperplane \( c^T x = c^T x^\star(t) \) is tangent to the level curve of \( \phi \) through \( x^\star(t) \).

- **Diagram:** A diagram illustrates the central path, showing a path from \( x^\star \) to \( x^\star(10) \), with a tangent line representing the hyperplane. The vector \( c \) is indicated, and level curves of \( \phi \) are shown as dashed lines.

**Source:** A. d'Aspremont. Convex Optimization M2. Slide 8/36.
```
