給定一個二次函數$f(x)=x^TAx+2a^Tx+a_0$，我們可以考慮由它所衍生出來的以下五個集合：$\{x\in\mathbb{R}^n |~ f(x) < 0\}$、$\{x\in\mathbb{R}^n |~ f(x) \leq 0\}$、$\{x\in\mathbb{R}^n |~ f(x) = 0\}$、$\{x\in\mathbb{R}^n |~ f(x) \geq 0\}$以及$\{x\in\mathbb{R}^n |~ f(x) > 0\}$。它們當中有些可能是非連通集，因此就有可能被另外一個二次函數的等高集$\{x\in\mathbb{R}^n |~ g(x)=x^TBx+2b^Tx+b_0=0\}$所切割。這樣的切割性質可以被用來研究聯合值域$\mathbf{R}(f,g) = \left\{\big(f(x), g(x)\big)~\big|~x\in \mathbb{R}^n \right\}$的形狀與位置，進而被用於解決二次規劃的問題。在本篇論文中，我們先研究集合$\{x\in\mathbb{R}^n |~ f(x) \star 0\}$是非連通集的充分必要條件，其中$\star \in \{<, \leq, =, \geq , >\}$。而後運用此組條件進一步刻劃所有滿足$\{x\in\mathbb{R}^n |~ g(x) = 0\}$切割非連通$\{x\in\mathbb{R}^n |~ f(x) \star 0\}$的二次函數$g(x)$。接著，我們利用兩個等高集的切割性質來刻畫聯合值域$\mathbf{R}(f,g) = \left\{\big(f(x), g(x)\big)~\big|~x\in \mathbb{R}^n \right\}$之凸性。最後，藉由重新檢視Farkas定理、傳統的$\mathcal{S}$-引理、以及等號版本的$\mathcal{S}$-引理，我們指出研究聯合值域的重要性。

Given a quadratic function $f(x)=x^TAx+2a^Tx+a_0$, it is possible that some of the five sets $\{x\in\mathbb{R}^n |~ f(x) < 0\}$, $\{x\in\mathbb{R}^n |~ f(x) \leq 0\}$, $\{x\in\mathbb{R}^n |~ f(x) = 0\}$, $\{x\in\mathbb{R}^n |~ f(x) \geq 0\}$, and $\{x\in\mathbb{R}^n |~ f(x) > 0\}$ are disconnected, and thus could be separated by another quadratic level set $\{x\in\mathbb{R}^n |~ g(x)=x^TBx+2b^Tx+b_0=0\}$. Such kind of separation property turns out to have several indications on the joint numerical range $\mathbf{R}(f,g) = \left\{\big(f(x), g(x)\big)~\big|~x\in \mathbb{R}^n \right\}$, and thus to have great implication in quadratic optimization. In this thesis, we start from the necessary and sufficient conditions for the set $\{x\in\mathbb{R}^n |~ f(x) \star 0\}$, $\star \in \{<, \leq, =, \geq , >\}$, to be disconnected. On top of that, we characterize all the possible quadratic function $g(x)$ whose $0$-level set $\{x\in\mathbb{R}^n |~ g(x)=0\}$ separates the disconnected $\{x\in\mathbb{R}^n |~ f(x) \star 0\}$. Then, we applied it to characterize the non-convexity of the joint numerical range $\mathbf{R}(f,g) = \left\{\big(f(x), g(x)\big)~\big|~x\in \mathbb{R}^n \right\}$. Finally, we point out the importance for the the shape and position of joint numerical range by reviewing Farkas Theorem, classical $\mathcal{S}$-lemma, and $\mathcal{S}$-lemma with equality.

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