Polarizable quasiconvex function generated metric projection
Abstract
We study when generalized metric projections generated by strongly quasiconvex functions are polarizable. The point of departure is Moreau's decomposition theorem, where the Euclidean metric projections onto two mutually polar closed convex cones form complementary retractions whose sum is the identity. This suggests the following fundamental question: when does a given retraction admit such a complementary polar retraction? We formulate this notion through polarizable retractions and then specialize it to generalized projections of the form x↦ Pcal Cφ(x), defined by minimizing φ(x−y) with respect to y over a closed convex set cal C. After recalling basic properties of mutually polar retractions and first-order characterizations of generalized projections, we prove the main result: If φ is strongly quasiconvex, the generalized projection Pcal Cφ is polarizable if and only if cal C is a closed convex cone. Thus, polarizability of the projection is characterized by the conic geometry of its range.
Related
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BibTeX
@misc{nemeth2026polarizable,
title = {{Polarizable quasiconvex function generated metric projection}},
author = {A. B. Németh and S. Z. Németh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22106},
eprint = {2610.22106},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22106},
}
RIS
TY - UNPB
TI - Polarizable quasiconvex function generated metric projection
AU - A. B. Németh
AU - S. Z. Németh
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22106
UR - https://arxiv.org/abs/2610.22106
AB - We study when generalized metric projections generated by strongly quasiconvex functions are polarizable. The point of departure is Moreau's decomposition theorem, where the Euclidean metric projections onto two mutually polar closed convex cones form complementary retractions whose sum is the identity. This suggests the following fundamental question: when does a given retraction admit such a complementary polar retraction? We formulate this notion through polarizable retractions and then specialize it to generalized projections of the form x↦ P_cal C^φ(x), defined by minimizing φ(x−y) with respect to y over a closed convex set cal C. After recalling basic properties of mutually polar retractions and first-order characterizations of generalized projections, we prove the main result: If φ is strongly quasiconvex, the generalized projection P_cal C^φ is polarizable if and only if cal C is a closed convex cone. Thus, polarizability of the projection is characterized by the conic geometry of its range.
ER -