On the convexity for the range set of two quadratic functions
Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu
TL;DR
This work addresses when the joint range $\igl( f(x), g(x) \bigr)$ of two quadratics with linear terms is convex. It recasts convexity as a geometric problem about separation of level sets and develops a polynomial-time algorithm that reduces to verifying $B=\lambda A$ and a hyperplane separation condition, expressed via a few linear-algebra tests. The key contributions are a necessary-and-sufficient separation criterion for non-convexity, a practical convexity-check procedure independent of SDP solves, and a unifying view that links to the $ ext{S}$-lemma and earlier results by Flores-Bazán & Opazo. This provides a principled, efficient tool for analyzing quadratic joint ranges in optimization and control contexts, with broader theoretical implications.
Abstract
Given $n\times n$ symmetric matrices $A$ and $B$, Dines in 1941 proved that the joint range set $\{(x^TAx,x^TBx)|~x\in\mathbb{R}^n\}$ is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set $\mathbf{R}(f,g) = \{\left(f(x),g(x)\right)|~x \in \mathbb{R}^n \},$ $f(x) = x^T A x + 2a^T x + a_0$ and $g(x) = x^T B x + 2b^T x + b_0.$ We show that $\mathbf{R}(f,g)$ is convex if, and only if, any pair of level sets, $\{x\in\mathbb{R}^n|f(x)=α\}$ and $\{x\in\mathbb{R}^n|g(x)=β\}$, do not separate each other. With the novel geometric concept about separation, we provide a polynomial-time procedure to practically check whether a given $\mathbf{R}(f,g)$ is convex or not.