Norm-Cone Conjugation for Nonconvex Optimization: Scalar Duality and Exact Distance Penalization
Abstract
We develop a norm-cone conjugation framework, generated by translated norm-cones x↦ r−α|x−x0|, α≥ 0. We show that this family generates the same abstract-convex class as the family of Lipschitz continuous concave functions, and exploit its explicit metric structure to obtain a concrete support geometry, a Fenchel--Moreau-type biconjugation theorem, and an associated norm-cone subdifferential. The main optimization consequence is a perturbation-duality theory based on partial norm-cone conjugation. Although the partial conjugate depends on a slope and a centre, the centre can be fixed at the nominal perturbation in the complete biconjugate expression without changing its value. Consequently, the effective dual problem is a scalar program whose only dual variable is the real slope α≥0, even when the primal and perturbation spaces are vector-valued or infinite-dimensional. The resulting dual value is the norm-cone biconjugate of the value function at the nominal perturbation, so that strong duality is characterized by exact pointwise biconjugation. For nonconvex conic inequality problems, exact distance penalization is characterized by norm-cone supportability of the natural value function: the exact penalty parameters form its norm-cone subdifferential, and the least exact parameter is the minimal support slope. Global error bounds, equivalently global metric subregularity, provide a variational-analytic sufficient condition for exactness and hence for strong norm-cone duality. An infinite-dimensional example shows that the results remain applicable with a nonconvex objective and a nonsolid ordering cone.
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BibTeX
@misc{garciacastano2026normcone,
title = {{Norm-Cone Conjugation for Nonconvex Optimization: Scalar Duality and Exact Distance Penalization}},
author = {Fernando García-Castaño and Miguel Ángel Melguizo-Padial},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.13495},
eprint = {2607.13495},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13495},
}
RIS
TY - UNPB
TI - Norm-Cone Conjugation for Nonconvex Optimization: Scalar Duality and Exact Distance Penalization
AU - Fernando García-Castaño
AU - Miguel Ángel Melguizo-Padial
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2607.13495
UR - https://arxiv.org/abs/2607.13495
AB - We develop a norm-cone conjugation framework, generated by translated norm-cones x↦ r−α|x−x_0|, α≥ 0. We show that this family generates the same abstract-convex class as the family of Lipschitz continuous concave functions, and exploit its explicit metric structure to obtain a concrete support geometry, a Fenchel--Moreau-type biconjugation theorem, and an associated norm-cone subdifferential. The main optimization consequence is a perturbation-duality theory based on partial norm-cone conjugation. Although the partial conjugate depends on a slope and a centre, the centre can be fixed at the nominal perturbation in the complete biconjugate expression without changing its value. Consequently, the effective dual problem is a scalar program whose only dual variable is the real slope α≥0, even when the primal and perturbation spaces are vector-valued or infinite-dimensional. The resulting dual value is the norm-cone biconjugate of the value function at the nominal perturbation, so that strong duality is characterized by exact pointwise biconjugation. For nonconvex conic inequality problems, exact distance penalization is characterized by norm-cone supportability of the natural value function: the exact penalty parameters form its norm-cone subdifferential, and the least exact parameter is the minimal support slope. Global error bounds, equivalently global metric subregularity, provide a variational-analytic sufficient condition for exactness and hence for strong norm-cone duality. An infinite-dimensional example shows that the results remain applicable with a nonconvex objective and a nonsolid ordering cone.
ER -