论文标题
关于一对二次函数的级别集的分离
On Separation of level sets for a pair of quadratic functions
论文作者
论文摘要
给定二次函数$ f（x）= x^tax+2a^tx+a_0，$，其级别设置$ \ {x \ in \ mathbb {r}^n：f（x）= 0 \} $具有两个连接的组件，因此可以按级别设置$ \ \ n \ n \ in \ in \ in \ n \ n \ n \ n \ n \ n In \ in \ n In \ in \ n \ n \ n \ n \ n \ n \ in \ n \ n In \ in \ n ar \ n \ n In \ in \ n ar \ n In \ n \ bb {在另一个二次函数$ g（x）= x^tbx+2b^tx+b_0。$事实证明，这种类型的分离属性对二次优化问题具有很大的含义，因此值得仔细的研究。在本文中，我们通过必要和充分的条件在分析上表征分离属性，作为解决优化问题的新工具。
Given a quadratic function $f(x)=x^TAx+2a^Tx+a_0,$ it is possible that its level set $\{x\in\mathbb{R}^n: f(x)=0\}$ has two connected components and thus can be separated by the level set $\{x\in\mathbb{R}^n: g(x)=0\}$ of another quadratic function $g(x)=x^TBx+2b^Tx+b_0.$ It turns out that the separation property of such kind has great implication in quadratic optimization problems and thus deserves careful studies. In this paper, we characterize the separation property analytically by necessary and sufficient conditions as a new tool to solving optimization problems.
