A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem
Van-Bong Nguyen, Huu-Quang Nguyen
TL;DR
The paper addresses the convexity and separability of the joint numerical range $G(R^n)$ for a special class of quadratic functions, linking these geometric properties to the smallest enclosing ball (SEB) problem. It proves a sharp rank-based criterion: $G(R^n)$ is convex iff ${\rm rank}\{a_1-a, \dots, a_m-a\} \le n-1$, and when equality holds with $m=n$, it introduces $G(R^n)^{\bullet}$ which remains convex and preserves separability with $\Lambda$. This separable structure enables an extended S-Lemma and a polynomial-time SDP reformulation of SEB, or equivalently a convex quadratic problem on the simplex, yielding practical algorithms for SEB under the identified regimes. The results delineate tractable versus potentially intractable instances and propose a conjecture on NP-hardness tied to the rank and size relation between the centers, with implications for outer-approximation methods and ellipsoidal containment problems.
Abstract
We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with smallest radius $r$ such that $B(a,r)$ contains the intersection $\cap_{i=1}^mB(a_i,r_i)$ of $m$ given balls $B(a_i,r_i).$ We show that $G(\Bbb R^n)$ is convex if and only if ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}\le n-1.$ Otherwise, ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}=n$ and $G(\Bbb R^n)$ is not convex. In this case we propose a new set $G(\Bbb R^n)^\bullet$ which allows to show that if $m=n$ then $G(\Bbb R^n)^\bullet$ is convex even $G(\Bbb R^n)$ is not. Importantly, the separable property of $G(\Bbb R^n)^\bullet$ then implies the separable property for $G(\Bbb R^n).$ As a result, a new progress on solving the SEB problem is obtained.