Details

Autor(en) / Beteiligte

• Nguyen, Huu-Quang
• Chu, Ya-Chi
• Sheu, Ruey-Lin

Titel

On the convexity for the range set of two quadratic functions

Ist Teil von

• Journal of industrial and management optimization, 2022-01, Vol.18 (1), p.575

Ort / Verlag

Springfield: American Institute of Mathematical Sciences

Erscheinungsjahr

2022

Link zum Volltext

Beschreibungen/Notizen

• Given <inline-formula><tex-math id="M1">\begin{document}$ n\times n $\end{document}</tex-math></inline-formula> symmetric matrices <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ B, $\end{document}</tex-math></inline-formula> Dines in 1941 proved that the joint range set <inline-formula><tex-math id="M4">\begin{document}$ \{(x^TAx, x^TBx)|\; x\in\mathbb{R}^n\} $\end{document}</tex-math></inline-formula> is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set <inline-formula><tex-math id="M5">\begin{document}$ \mathbf{R}(f, g) = \{\left(f(x), g(x)\right)|\; x \in \mathbb{R}^n \}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M6">\begin{document}$ f(x) = x^T A x + 2a^T x + a_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ g(x) = x^T B x + 2b^T x + b_0. $\end{document}</tex-math></inline-formula> We show that <inline-formula><tex-math id="M8">\begin{document}$ \mathbf{R}(f, g) $\end{document}</tex-math></inline-formula> is convex if, and only if, any pair of level sets, <inline-formula><tex-math id="M9">\begin{document}$ \{x\in\mathbb{R}^n|f(x) = \alpha\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \{x\in\mathbb{R}^n|g(x) = \beta\} $\end{document}</tex-math></inline-formula>, do not separate each other. With the novel geometric concept about separation, we provide a polynomial-time procedure to practically check whether a given <inline-formula><tex-math id="M11">\begin{document}$ \mathbf{R}(f, g) $\end{document}</tex-math></inline-formula> is convex or not.