```markdown
### Implicit Constraints

The slide discusses the concept of implicit constraints in optimization problems. It includes the following points:

- **Implicit Constraint Definition**: The standard form optimization problem has an implicit constraint defined as:
  \[
  x \in \mathcal{D} = \bigcap_{i=0}^m \text{dom } f_i \cap \bigcap_{i=1}^p \text{dom } h_i
  \]
  
- **Domain**: \(\mathcal{D}\) is referred to as the domain of the problem.

- **Explicit Constraints**: The constraints \(f_i(x) \leq 0\) and \(h_i(x) = 0\) are considered explicit constraints.

- **Unconstrained Problems**: A problem is unconstrained if it has no explicit constraints (\(m = p = 0\)).

#### Example:

- The function to minimize is:
  \[
  f_0(x) = -\sum_{i=1}^k \log(b_i - a_i^T x)
  \]

- This is an unconstrained problem with implicit constraints \(a_i^T x < b_i\).

---

**Source**: A. d'Aspremont, Convex Optimization M2, Slide 37/67
```
