```markdown
### Perturbation and Sensitivity Analysis

#### (Unperturbed) Optimization Problem and Its Dual

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

- **Maximize:**
  \[
  g(\lambda, \nu)
  \]
  **Subject to:**
  \[
  \lambda \succeq 0
  \]

#### Perturbed Problem and Its Dual

- **Minimize:**
  \[
  f_0(x)
  \]
  **Subject to:**
  \[
  f_i(x) \leq u_i, \quad i = 1, \ldots, m
  \]
  \[
  h_i(x) = v_i, \quad i = 1, \ldots, p
  \]

- **Maximize:**
  \[
  g(\lambda, \nu) - u^T \lambda - v^T \nu
  \]
  **Subject to:**
  \[
  \lambda \succeq 0
  \]

#### Notes

- \( x \) is the primal variable; \( u, v \) are parameters.
- \( p^*(u, v) \) is the optimal value as a function of \( u, v \).
- We are interested in information about \( p^*(u, v) \) that can be obtained from the solution of the unperturbed problem and its dual.

---

A. d’Aspremont. Convex Optimization M2.  
24/49
```
