```markdown
### KKT conditions for convex problem

If \( \tilde{x}, \tilde{\lambda}, \tilde{\nu} \) satisfy KKT for a **convex problem**, then they are optimal:

- From complementary slackness: \( f_0(\tilde{x}) = L(\tilde{x}, \tilde{\lambda}, \tilde{\nu}) \)
- From 4th condition (and convexity): \( g(\tilde{\lambda}, \tilde{\nu}) = L(\tilde{x}, \tilde{\lambda}, \tilde{\nu}) \)

Hence, \( f_0(\tilde{x}) = g(\tilde{\lambda}, \tilde{\nu}) \).

If **Slater’s condition** is satisfied, \( x \) is optimal if and only if there exist \( \lambda, \nu \) that satisfy KKT conditions:

- Recall that Slater implies strong duality, and dual optimum is attained.
- Generalizes optimality condition \( \nabla f_0(x) = 0 \) for unconstrained problem.

### Summary:

- When strong duality holds, the KKT conditions are necessary conditions for optimality.
- If the problem is **convex**, they are also sufficient.

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