Calmness Modulus of Parameterized Convex Constraint Systems: Exact Closed-Form Formulas via Second-Order Admissibility

Abstract

We resolve the open problem of deriving an exact, closed-form algebraic formula for the calmness modulus of ${\mathcal{C}}^2$ convex constrained feasible set mappings $\mathcal{F}(b):=\{x\in {\mathbb{R}}^n:f_t(x)\le b_t,\forall t\in T\}$ under right-hand side perturbations at a given pair of the graph $(\bar{b},\bar{x})$. First, under the condition $0_n\in \mathrm{int}\left(\mathrm{conv}\left\{\nabla f_t(\bar{x}),t\in T(\bar{x})\right\}\right)$, we prove that the calmness modulus coincides with the inverse distance from the origin to the boundary of the nominal subdifferential. Second, for general ${\mathcal{C}}^2$ convex systems where the previously mentioned interior condition does not hold, we introduce a second-order admissibility filtering over the active indexes. We show that this filtered family, ${\mathcal{D}}_2^0(\bar{x})$, yields an exact characterization of the calmness modulus relying solely on nominal data.