Convex Optimization 2026, 18 - Perturbation and Sensitivity

Strong duality has optimal dual variables yield information about sensitivity of the optimal value wrt. to perturbations of the constraints. A perturbed problem is a normal problem with relaxed or tightened constraints. At null parameters, the optimal values should coincide (5.6.1). For strong duality with dual optimum, and optimal (λ*, ν*), then ∀ u, v: p*(u, v) ≥ p*(0, 0) - λ*^Tu - ν*^Tv (5.6.2).

If λ*_i is large and the ith constraint is tightened, then the optimal value will increase greatly. If ν* is large and positive, and v_i < 0, of if ν*_i is large and negative, and v_i > 0, then the optimal value increases greatly (5.6.2). Given opposite conditions, the optimal value won't decrease too much. If the optimal value is differentiable at u = 0, v = 0, with strong duality, the optimal dual variables are related of the gradient for the optimal value at origin. It follows that λ*_i = -∂_uip*(0, 0); and ν*_i = -∂_vip*(0, 0) (Eq. 5.58).