Quadratic Convexification of a Square Truncated by a Hyperbola

Abstract

We study the quadratic convexification of the compact nonconvex set $G:={\left[1/2,2\right]}^2\cap \{(x_1,x_2)\in {\mathbb{R}}^2\mid x_1{x}_2\le 1\},$ 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.