```markdown
### Descent Methods

**Equation:**
\[ x^{(k+1)} = x^{(k)} + t^{(k)} \Delta x^{(k)} \]
with \( f(x^{(k+1)}) < f(x^{(k)}) \)

**Notations:**
- \( x^+ = x + t \Delta x \), \( x := x + t \Delta x \)
- \(\Delta x\) is the step, or search direction; \( t \) is the step size, or step length.

**Convexity:**
- If \( f(x^+) < f(x) \), it implies \( \nabla f(x)^T \Delta x < 0 \) (i.e., \(\Delta x\) is a descent direction).

**General Descent Method:**

Given a starting point \( x \in \text{dom } f \).

**Repeat:**
1. Determine a descent direction \(\Delta x\).
2. Line search. Choose a step size \( t > 0 \).
3. Update: \( x := x + t \Delta x \).

**Until** stopping criterion is satisfied.

*Source: A. d’Aspremont. Convex Optimization M2.*
```
