论文标题
平面凸子集的二阶锥体表示
Second-order cone representation for convex subsets of the plane
论文作者
论文摘要
半决赛编程（SDP）是在有限的许多线性矩阵不等式（LMIS）的公共解决方案集上优化线性函数的任务。对于SDP求解器的运行时间，这些LMI的最大矩阵大小通常比其数字更为重要。凸的半芬矿扩展度$ \ text {sxdeg}（k）$ $ k \ subseteq \ subseteq \ mathbb r^n $是最小的数字$ d $ $ d $，因此$ k $是有限的相互作用的线性图像$ s_1 \ s_1 \ cap \ cap \ cap \ cap s_n $，每个$ s_n $ s a $ s_i ah $ s_i ah in Is a a s a a seppectrii s a a a a a a a a seppece a a a a a a a seppece a a a seppece a a a seppectriix大小$ \ le d $。因此，$ \ text {sxdeg}（k）$可以看作是在集合$ k $上执行半决赛程序的复杂性的措施。我们给出了$ \ text {sxdeg}（k）$的几种等效特征，并使用它们来证明我们的主要结果：$ \ text {sxdeg}（k）\ le2 $均可在任何封闭的凸semialgebraic set $ k \ subseteq \ subseteq \ mathbb r^2 $中保留。换句话说，可以使用二阶锥体代表这种$ K $。
Semidefinite programming (SDP) is the task of optimizing a linear function over the common solution set of finitely many linear matrix inequalities (LMIs). For the running time of SDP solvers, the maximal matrix size of these LMIs is usually more critical than their number. The semidefinite extension degree $\text{sxdeg}(K)$ of a convex set $K\subseteq\mathbb R^n$ is the smallest number $d$ such that $K$ is a linear image of a finite intersection $S_1\cap\dots\cap S_N$, where each $S_i$ is a spectrahedron defined by a linear matrix inequality of size $\le d$. Thus $\text{sxdeg}(K)$ can be seen as a measure for the complexity of performing semidefinite programs over the set $K$. We give several equivalent characterizations of $\text{sxdeg}(K)$, and use them to prove our main result: $\text{sxdeg}(K)\le2$ holds for any closed convex semialgebraic set $K\subseteq\mathbb R^2$. In other words, such $K$ can be represented using the second-order cone.
