带双线性约束的象限上的非负二次型
Nonnegative Quadratics over a Quadrant with a Bilinear Constraint
AI总结：
本文刻画带双线性约束的象限上非负二次多项式锥的极射线，得到其提升凸包的半定表示，还构造了平方和锥的六维线性截面并给出两类多项式的有界次非负性证书。
AI中文摘要：
我们研究在非紧集 \\( F:=\{(x_1,x_2)\in\mathbb R^2:\\ x_1\ge 0,\\ x_2\ge 0,\\ x_1x_2\le 1\} \\) 上非负的二次多项式，刻画该集合上非负二次多项式锥的所有极射线。该刻画使我们能研究涉及 \\( F \\) 的二次凸化的参数化有效不等式，得到其在二次空间中提升凸包的半定表示。我们对极射线刻画的分析将半正定（PSD）分支与非半正定分支分离，将后者简化为边界非负性，并对相关极射线的边界接触进行分类。通过重新参数化，我们还找到非平凡且非排列对称的六维线性截面，该截面属于非负三元八次多项式的平方和锥。此外，我们证明提升凸包结果给出 \\( F \\) 上非负二次型的有界次预序证书，以及半带上一类非负四次多项式的有界次证书。
英文摘要：
We study quadratic polynomials that are nonnegative on the non-compact set \[ F:=\{(x_1,x_2)\in\mathbb R^2:\ x_1\ge 0,\ x_2\ge 0,\ x_1x_2\le 1\}. \] All extreme rays of the cone of nonnegative quadratic polynomials are characterized on this set. The characterization allows us to study parameterized valid inequalities for quadratic convexifications involving $F$, which yields a semidefinite representation of its lifted convex hull in the quadratic space. Our analysis on extreme ray characterization separates the positive-semidefinite (PSD) and non-PSD branches, reduces the latter to boundary nonnegativity, and classifies the boundary contacts of the relevant extreme rays. By reparameterization, we also find a non-trivial and non-permutation-symmetric six-dimensional linear section of the cone of nonnegative homogeneous ternary octics that are sum-of-squares. We also show that our lifted convex hull result yields a degree-bounded preordering certificate of nonnegative quadratics on $F$ and a degree-bounded certificate for a family of nonnegative quartics on the half-strip.
