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 𝒞2 convex constrained feasible set mappings ℱ(b) := { x ∈ ℝn : ft(x) ≤ bt, ∀ t ∈ T } under right-hand side perturbations at a given pair of the graph (b̅, x̅). First, under the condition 0n ∈ int(conv{ ∇ ft(x̅) , t ∈ T(x̅) }), we prove that the calmness modulus coincides with the inverse distance from the origin to the boundary of the nominal subdifferential. Second, for general 𝒞2 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, 𝒟02(x̅), yields an exact characterization of the calmness modulus relying solely on nominal data.
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BibTeX
@misc{camacho2026calmness,
title = {{Calmness Modulus of Parameterized Convex Constraint Systems: Exact Closed-Form Formulas via Second-Order Admissibility}},
author = {Jesús Camacho},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06224},
eprint = {2611.06224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06224},
}
RIS
TY - UNPB
TI - Calmness Modulus of Parameterized Convex Constraint Systems: Exact Closed-Form Formulas via Second-Order Admissibility
AU - Jesús Camacho
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06224
UR - https://arxiv.org/abs/2611.06224
AB - We resolve the open problem of deriving an exact, closed-form algebraic formula for the calmness modulus of 𝒞^2 convex constrained feasible set mappings ℱ(b) := { x ∈ ℝ^n : f_t(x) ≤ b_t, ∀ t ∈ T } under right-hand side perturbations at a given pair of the graph (b̅, x̅). First, under the condition 0_n ∈ int(conv{ ∇ f_t(x̅) , t ∈ T(x̅) }), we prove that the calmness modulus coincides with the inverse distance from the origin to the boundary of the nominal subdifferential. Second, for general 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, 𝒟^0_2(x̅), yields an exact characterization of the calmness modulus relying solely on nominal data.
ER -