設 f(x) 和 g_i(x), i = 1, 2, . . . , m 為 n 元二次實係數多項式，並將二次約束二次規劃問題 (QPQC) 定義作最小化 f (x) 於限制集合 {x | g(x) ∗ 0, x ∈ X} 上，其中 ∗ ∈ {<, ≤, =} 且 X = R^n 或 X = S^(n−1) 。在 {x | g_i(x) < 0} 的情形下，也就是限制式會有嚴格不等式的情況，其結果在文獻上實屬罕見。在本論文中，我們建立一個一般性的框架來處理 (QPQC)，且其中包含了 ∗ ∈ {<, ≤, =} 以及 X = R^n 或 X = S^(n−1) 的所有情況。在此框架下新定義「多重 Slater 條件」並在聯合數值域 {f (x), g_1(x), . . . , g_m(x) | x ∈ X} 是凸的假設下，可以得到 (QPQC) 的強對偶定理。本論文有助於我們理解 S-引理、等號形式之 S-引理及 Finsler 引理，此外也給出了非齊次 Finsler 引理的可能研究方向。

Let f(x) and g_i(x), i = 1, 2, . . . , m be n-variate real-valued quadratic functions and define the quadratic programming with quadratic constraints (QPQC) as minimizing f(x) over {x | g(x) ∗ 0, x ∈ X}, where ∗ ∈ {<, ≤, =} and X = R^n or X = S^(n−1). In literature, the case when {x | g_i(x) < 0} for some i, i.e. the constraint set consists of some strict inequalities, has rarely been discussed. By this thesis, we build a general framework to handle (QPQC) with all ∗ ∈ {<, ≤, =} and X = R^n or X = S^(n−1) , within which the strong duality of (QPQC) is established under a newly developed “multiple Slater condition” and the assumption that the joint numerical range {f (x), g_i (x), . . . , g_m (x) | x ∈ X} is convex. Our result is helpful for us to understand S-lemma, S-lemma with equality, and Finsler lemma, and give a direction of research on non-homogenized Finsler lemma.

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