給定一個目標函數和限制式皆為二次凸函數，f_i(x)=x^TQ_ix+2b_i^Tx+beta_i, Q_i為半正定, i=0,1,...,m 的二次規劃問題，我們的主要工作是判斷目標函數在可行解區域是無界還是有界。這個問題在1985年Terlaky將它轉換成l_p規劃得以解決，並在1995年由Caron和Obuchowska給出一個演算法來判斷，最後在2020年Nguyen和Sheu給出它在數學上的充分且必要條件。在本篇論文中，我們將證明Nguyen和Sheu在數學上的充分且必要條件是等價於Caron和Obuchowska的演算法。接著我們以Nguyen和Sheu的充分且必要條件為基礎，給出一個新的判斷本問題無界的演算法，其計算效率或許僅和Caron和Obuchowska的演算法相當，然而計算步驟會較清楚簡潔，並且有辦法給出一系列的曲線，使得目標函數沿著這些曲線可以無界遞減。在本篇論文中我們也給出許多例子來輔助說明。

Given a convex quadratic objective function and convex quadratic constraints, f_i(x)=x^TQ_ix+2b_i^Tx+beta_i, Q_i is positive semi-definite, i=0,1,...,m, our main task is to determine whether the objective function is unbounded from below in the feasible region. This problem was first solved in 1985 by Terlaky who transformed it into an l_p programming. Later in 1995, Caron and Obuchowska gave an algorithm for it. Finally, in 2020, Nguyen and Sheu gave a necessary and sufficient condition for it. In our thesis, we prove that Nguyen and Sheu's necessary and sufficient condition is equivalent to Caron and Obuchowska's algorithm. Based on the necessary and sufficient condition of Nguyen and Sheu is proposed a new algorithm to decide the unboundedness of the convex quadratic problem. Although the computational efficiency of the new algorithm may only be roughly the same as that of Caron and Obuchowska's, yet the iterative process of the new proposed algorithm is more clear and concise. More Valuably, the new method can construct curves along which the objective function decreases without bound. Many illustrative examples are inserted for better explaining the idea.

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