Stability and Robustness: Reliability in the World of Uncertainty

Sketch of the proofProblemFormulationStability andAccuracyFunctionsSo, we have that S(C 0 , J, x ∗ , ρ) ≤ Γ(C 0 , J, x ∗ , ρ). Now itremains to show that S(C 0 , J, x ∗ , ρ) ≥ Γ(C 0 , J, x ∗ , ρ). Considera matrix C ∗ with elements defined according to (2) for anyr ∈ N s . ThenS(C 0 , x ∗ , J, ρ) =maxC∈Ω ρ (C 0 ) ε(C, J, x∗ ) ≥ ε(C ∗ , J, x ∗ ) =maxr∈N smaxr∈N smaxx∈W J r (x∗ ) mini∈J rC ∗ i (x∗ − x)C ∗ i x =max min Ci 0(x∗ − x) + ρ ‖ x ∗ − x ‖ 1x∈W J r (x∗ ) i∈J r Ci 0x − ρ ‖ x ‖ 1So, we have that S(C 0 , x ∗ , J, ρ) ≥ Γ(C 0 , x ∗ , J, ρ). Thiscompletes the proof.= Γ(C 0 , J, x ∗ , ρ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 20/29