On error bounds and noncritical Lagrange multipliers
Abstract
Critical Lagrange multipliers are known to be responsible for the potentially slow convergence of Newton-type methods, and noncriticality of a given multiplier has been tied to an error bound which estimates the distance of a primal-dual pair associated with a perturbed problem to the primal solution and the multiplier set of the unperturbed one, and is decisive for a local convergence analysis. We investigate this relation in an abstract setting where merely a set-valued mapping assigning primal-dual pairs to a parameter is available, and characterize the error bound of interest in terms of noncriticality of the underlying multiplier and two calmness-type properties of the associated multiplier mapping, one of which is a novel weak inner calmness in the fuzzy sense. No structural assumptions on the underlying problem are needed for that, and the two calmness-type conditions are not only sufficient but also necessary. Afterwards, we specify these findings for composite optimization problems. Whenever the subdifferential of the outer function is a polyhedral mapping, both conditions hold automatically, which explains the equivalence of noncriticality and the error bound observed in the polyhedral setting. Whenever the outer function is C2-decomposable, they are secured by calmness of a restricted multiplier mapping together with a closedness condition, which recovers and extends the known characterization for C2-cone reducible constraint systems.
Related
- Degeneracies of Lagrange functions and geometric interpretations of Lagrange multipliers
- On the Choice of Continuous and Discontinuous Lagrange Multipliers in Fictitious-Domain Methods
- Dual companions and improved regularity of Lagrange multipliers in optimal control
- A priori error analysis for one-parameter Lagrange and POD reduced order methods
- Sharp Probabilistic Normwise Backward Error Bounds for LU Factorization
BibTeX
@misc{benko2026error,
title = {{On error bounds and noncritical Lagrange multipliers}},
author = {Matúš Benko and Patrick Mehlitz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11491},
eprint = {2610.11491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11491},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On error bounds and noncritical Lagrange multipliers
AU - Matúš Benko
AU - Patrick Mehlitz
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.11491
UR - https://arxiv.org/abs/2610.11491
AB - Critical Lagrange multipliers are known to be responsible for the potentially slow convergence of Newton-type methods, and noncriticality of a given multiplier has been tied to an error bound which estimates the distance of a primal-dual pair associated with a perturbed problem to the primal solution and the multiplier set of the unperturbed one, and is decisive for a local convergence analysis. We investigate this relation in an abstract setting where merely a set-valued mapping assigning primal-dual pairs to a parameter is available, and characterize the error bound of interest in terms of noncriticality of the underlying multiplier and two calmness-type properties of the associated multiplier mapping, one of which is a novel weak inner calmness in the fuzzy sense. No structural assumptions on the underlying problem are needed for that, and the two calmness-type conditions are not only sufficient but also necessary. Afterwards, we specify these findings for composite optimization problems. Whenever the subdifferential of the outer function is a polyhedral mapping, both conditions hold automatically, which explains the equivalence of noncriticality and the error bound observed in the polyhedral setting. Whenever the outer function is C^2-decomposable, they are secured by calmness of a restricted multiplier mapping together with a closedness condition, which recovers and extends the known characterization for C^2-cone reducible constraint systems.
ER -