```markdown
### Lagrangian

**Standard form problem** (not necessarily convex)

- Minimize \( f_0(x) \)
- Subject to \( f_i(x) \leq 0 \), \( i = 1, \ldots, m \)
- \( h_i(x) = 0 \), \( i = 1, \ldots, p \)

Variable \( x \in \mathbb{R}^n \), domain \(\mathcal{D}\), optimal value \( p^\star \).

**Lagrangian:** \( L : \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow \mathbb{R} \), with \(\text{dom } L = \mathcal{D} \times \mathbb{R}^m \times \mathbb{R}^p\),

\[
L(x, \lambda, \nu) = f_0(x) + \sum_{i=1}^{m} \lambda_i f_i(x) + \sum_{i=1}^{p} \nu_i h_i(x)
\]

- Weighted sum of objective and constraint functions
- \(\lambda_i\) is Lagrange multiplier associated with \( f_i(x) \leq 0 \)
- \(\nu_i\) is Lagrange multiplier associated with \( h_i(x) = 0 \)

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