由双曲线截断的正方形的二次凸化
Quadratic Convexification of a Square Truncated by a Hyperbola
AI总结：
研究由双曲线截断的正方形构成的非凸集$G$，刻画其非负二次多项式锥的极射线，推导其提升凸包的精确有限半定表示，完善了乘积约束区域的凸化理论。
AI中文摘要：
我们研究紧非凸集$G:=[1/2,2]^2 \cap \{(x_1,x_2)\in\mathbb R^2\mid x_1x_2\leq1\}$的二次凸化，该集合是由双曲线弧截断的正方形。尽管$G$的普通凸包是一个三角形，但它在完全二次空间中的提升凸包保留了弯曲边界的非平凡几何结构。我们刻画了在$G$上非负的二次多项式锥的所有极射线，包括有界切线和双切线射线的参数化族。利用该分类和锥对偶性，我们推导了$G$的提升凸包的精确有限半定表示。该分析遵循我们早期关于无界乘积约束区域的工作的一般框架，但有界几何产生了新的边界接触模式，并导致非负二次锥的不同有限组织。
英文摘要：
We study the quadratic convexification of the compact nonconvex set ${G} := \left[1/2,2\right]^2 \cap \{(x_1,x_2)\in\mathbb R^2\mid x_1x_2\leq1\}, $ which is a square truncated by a hyperbolic arc. Although the ordinary convex hull of ${G}$ is a triangle, its lifted convex hull in the complete quadratic space retains the nontrivial geometry of the curved boundary. We characterize all extreme rays of the cone of quadratic polynomials nonnegative on $G$, including parameterized families of bounded tangent and bitangent rays. Using this classification and conic duality, we derive an exact finite semidefinite representation of the lifted convex hull of ${G}$. The analysis follows the general framework of our earlier work on an unbounded product-constrained region, but the bounded geometry creates new boundary-contact patterns and leads to a different finite organization of the nonnegative-quadratic cone.
