Why Study the Spherical Convexity of Non-Homogeneous Quadratic Functions, and What Makes It Surprising?
R. Bolton, S. Z. Németh
TL;DR
The paper addresses spherical convexity of non-homogeneous quadratic functions on the sphere and its proper spherically convex subsets, contrasting the sphere with hyperbolic and Euclidean settings. It develops necessary, sufficient, and equivalent conditions for spherical convexity of $f_{A,b,c}(x)=x^T A x + b^T x + c$ on sets $\mathbb{S}^n \cap \operatorname{int}(K)$, notably introducing the $E(A,b,c,u,v)$ criterion and gradient/Hessian-based characterizations. It shows that the class of spherically convex non-homogeneous quadratics is large—every symmetric $A$ admits infinitely many $b$ making $f$ spherically convex—and provides explicit, checkable certificates for special matrix types (positive, diagonal, $Z$-matrices) via copositivity and related conditions. The results have implications for optimization on manifolds, including trust-region methods and seismic inversion, and pave the way for future work on broader spherically convex domains and more general matrices, such as Lorentz-cone intersections with the sphere.
Abstract
This paper presents necessary, sufficient, and equivalent conditions for the spherical convexity of non-homogeneous quadratic functions. In addition to motivating this study and identifying useful criteria for determining whether such functions are spherically convex, we discovered surprising properties that distinguish spherically convex quadratic functions from their geodesically convex counterparts in both hyperbolic and Euclidean spaces. Since spherically convex functions over the entire sphere are constant, we restricted our focus to proper spherically convex subsets of the sphere. Although most of our results pertain to non-homogeneous quadratic functions on the spherically convex set of unit vectors with positive coordinates, we also present findings for more general spherically convex sets. Beyond the general non-homogeneous quadratic functions, we consider explicit special cases where the matrix in the function's definition is of a specific type, such as positive, diagonal, and Z-matrix.