Necessary Optimality Conditions for Bilevel Programming using Tangential Subdifferentials
Abstract
The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian--Fromovitz constraint qualification (MFCQ). In this paper, we study necessary optimality conditions for nonsmooth bilevel programming using tangential subdifferentials. We derive a sequential enhanced tangential Fritz--John type condition for a class of nonsmooth nonlinear programs and then obtain corresponding enhanced tangential Fritz--John and enhanced tangential Karush--Kuhn--Tucker (KKT) systems by taking limits of the sequential condition. We further investigate tangential constraint qualification conditions, including T-quasinormality and a tangential cone--continuity property (T--CCP), and analyze how they guarantee the validity of tangential KKT type conditions. Finally, these results are applied to the value-function reformulation of a bilevel programming problem.
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BibTeX
@misc{luo2026necessary,
title = {{Necessary Optimality Conditions for Bilevel Programming using Tangential Subdifferentials}},
author = {Huilin Luo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36948},
eprint = {2609.36948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36948},
}
RIS
TY - UNPB
TI - Necessary Optimality Conditions for Bilevel Programming using Tangential Subdifferentials
AU - Huilin Luo
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36948
UR - https://arxiv.org/abs/2609.36948
AB - The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian--Fromovitz constraint qualification (MFCQ). In this paper, we study necessary optimality conditions for nonsmooth bilevel programming using tangential subdifferentials. We derive a sequential enhanced tangential Fritz--John type condition for a class of nonsmooth nonlinear programs and then obtain corresponding enhanced tangential Fritz--John and enhanced tangential Karush--Kuhn--Tucker (KKT) systems by taking limits of the sequential condition. We further investigate tangential constraint qualification conditions, including T-quasinormality and a tangential cone--continuity property (T--CCP), and analyze how they guarantee the validity of tangential KKT type conditions. Finally, these results are applied to the value-function reformulation of a bilevel programming problem.
ER -