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## Perturbation and Sensitivity Analysis

**Global Sensitivity Result:**  
Strong duality holds for the unperturbed problem, and \(\lambda^\star, \nu^\star\) are dual optimal for the unperturbed problem. Apply **weak duality** to the perturbed problem:

\[
p^\star(u, v) \geq g(\lambda^\star, \nu^\star) - u^T \lambda^\star - v^T \nu^\star = p^\star(0, 0) - u^T \lambda^\star - v^T \nu^\star
\]

**Local Sensitivity:**  
If (in addition) \(p^\star(u, v)\) is differentiable at \((0, 0)\), then

\[
\lambda^\star_i = -\frac{\partial p^\star(0, 0)}{\partial u_i}, \quad \nu^\star_i = -\frac{\partial p^\star(0, 0)}{\partial v_i}
\]

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*A. d'Aspremont. Convex Optimization M2.*  
Page 25/49
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