On Separation of level sets for a pair of quadratic functions

TL;DR

This paper characterizes when two quadratic functions' level sets can be separated by another quadratic level set, providing analytical conditions that aid in solving quadratic optimization problems.

Contribution

It introduces necessary and sufficient conditions for the separation of quadratic level sets, advancing the understanding of their geometric and optimization implications.

Findings

Provides analytical characterization of separation conditions.

Links separation properties to quadratic optimization problems.

Offers a new tool for solving quadratic optimization problems.

Abstract

Given a quadratic function $f(x)=x^TAx+2a^Tx+a_0,$ it is possible that its level set $\{x\in\mathbb{R}^n: f(x)=0\}$ has two connected components and thus can be separated by the level set $\{x\in\mathbb{R}^n: g(x)=0\}$ of another quadratic function $g(x)=x^TBx+2b^Tx+b_0.$ It turns out that the separation property of such kind has great implication in quadratic optimization problems and thus deserves careful studies. In this paper, we characterize the separation property analytically by necessary and sufficient conditions as a new tool to solving optimization problems.