On the convexity for the range set of two quadratic functions

Abstract

Given $n\times n$ symmetric matrices $A$ and $B$, Dines in 1941 proved that the joint range set $\{(x^{T}Ax,x^{T}Bx)|x\in {\mathbb{R}}^n\}$ is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set $\mathbf{R}(f,g)=\{\left(f(x),g(x)\right)|x\in {\mathbb{R}}^n\},$ $f(x)=x^{T}Ax+2a^{T}x+a_0$ and $g(x)=x^{T}Bx+2b^{T}x+b_0.$ We show that $\mathbf{R}(f,g)$ is convex if, and only if, any pair of level sets, $\{x\in {\mathbb{R}}^n|f(x)=\alpha \}$ and $\{x\in {\mathbb{R}}^n|g(x)=\beta \}$, 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.

References in corpus (1)

• S-Lemma with Equality and Its Applications