Hidden Convexity of the Joint Range of Three Quadratic Forms and Applications
Abstract
We investigate a hidden convexity phenomenon for triples of homogeneous quadratic forms. We give a complete characterization of the convexity of the upper image obtained by adding the nonnegative orthant to their joint range. More precisely, the upper image is convex if and only if either a nonzero nonnegative linear combination of the three associated matrices is positive semidefinite, or the three quadratic forms admit a common strict negative direction. In particular, the upper image is convex whenever one of the three quadratic forms is convex. As applications, we establish the convexity of an associated value function and derive a generalized S-lemma for implications involving two quadratic inequalities and a strict theorem of the alternative for quadratic systems.
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BibTeX
@misc{hoang2026hidden,
title = {{Hidden Convexity of the Joint Range of Three Quadratic Forms and Applications}},
author = {Tuan Ngoc Hoang and Huy Quang Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30706},
eprint = {2610.30706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30706},
}
RIS
TY - UNPB
TI - Hidden Convexity of the Joint Range of Three Quadratic Forms and Applications
AU - Tuan Ngoc Hoang
AU - Huy Quang Nguyen
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30706
UR - https://arxiv.org/abs/2610.30706
AB - We investigate a hidden convexity phenomenon for triples of homogeneous quadratic forms. We give a complete characterization of the convexity of the upper image obtained by adding the nonnegative orthant to their joint range. More precisely, the upper image is convex if and only if either a nonzero nonnegative linear combination of the three associated matrices is positive semidefinite, or the three quadratic forms admit a common strict negative direction. In particular, the upper image is convex whenever one of the three quadratic forms is convex. As applications, we establish the convexity of an associated value function and derive a generalized S-lemma for implications involving two quadratic inequalities and a strict theorem of the alternative for quadratic systems.
ER -