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arXiv · 2610.30706

Hidden Convexity of the Joint Range of Three Quadratic Forms and Applications

Tuan Ngoc Hoang·Huy Quang Nguyen
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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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Tuan Ngoc Hoang, Huy Quang Nguyen. 2026-10-22. Hidden Convexity of the Joint Range of Three Quadratic Forms and Applications. https://arxiv.org/abs/2610.30706

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