```markdown
### Gradient Descent Method

**General Descent Method:**
- Update step: \(\Delta x = -\nabla f(x)\)

**Procedure:**
1. **Given** a starting point \(x \in \text{dom } f\).
2. **Repeat:**
   1. Compute \(\Delta x := -\nabla f(x)\).
   2. Perform a line search to choose step size \(t\) via exact or backtracking line search.
   3. Update: \(x := x + t\Delta x\).
3. **Until** the stopping criterion is satisfied.

**Key Points:**
- **Stopping Criterion:** Typically \(\|\nabla f(x)\|_2 \leq \epsilon\).
- **Convergence Result:** For strongly convex \(f\),
  \[
  f(x^{(k)}) - p^\star \leq c^k(f(x^{(0)}) - p^\star)
  \]
  where \(c \in (0, 1)\) depends on \(m\), \(x^{(0)}\), and line search type.
- **Note:** The method is very simple but often very slow; rarely used in practice.

**Source:** A. d'Aspremont, Convex Optimization M2.
```
