```markdown
## Complementary Slackness

Assume strong duality holds, \( x^* \) is primal optimal, \((\lambda^*, \nu^*)\) is dual optimal.

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

\[
\leq f_0(x^*) + \sum_{i=1}^{m} \lambda_i^* f_i(x^*) + \sum_{i=1}^{p} \nu_i^* h_i(x^*) 
\]

\[
\leq f_0(x^*) 
\]

Hence, the two inequalities hold with equality.

- \( x^* \) minimizes \( L(x, \lambda^*, \nu^*) \)
- \( \lambda_i^* f_i(x^*) = 0 \) for \( i = 1, \ldots, m \) (known as complementary slackness):
  - \( \lambda_i^* > 0 \implies f_i(x^*) = 0 \)
  - \( f_i(x^*) < 0 \implies \lambda_i^* = 0 \)

*Source:* A. d’Aspremont. Convex Optimization M2. Slide 20/49
```
