```markdown
### Convex Optimization Problem

**Standard Form Convex Optimization Problem:**

- **Objective:** Minimize \( f_0(x) \)
- **Subject to:**
  - \( f_i(x) \leq 0, \quad i = 1, \ldots, m \)
  - \( a_i^T x = b_i, \quad i = 1, \ldots, p \)

#### Key Points:
- \( f_0, f_1, \ldots, f_m \) are convex; equality constraints are affine.
- The problem is **quasiconvex** if \( f_0 \) is quasiconvex (and \( f_1, \ldots, f_m \) are convex).

#### Alternate Formulation:
- **Objective:** Minimize \( f_0(x) \)
- **Subject to:**
  - \( f_i(x) \leq 0, \quad i = 1, \ldots, m \)
  - \( Ax = b \)

#### Important Property:
- The feasible set of a convex optimization problem is convex.

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