Geometric properties for level sets of quadratic functions

Abstract

In this paper, we study some fundamental geometrical properties related to the ${\mathcal {S}}$-procedure. Given a pair of quadratic functions (g, f), it asks when “$g(x)=0 \Longrightarrow ~ f(x)\ge 0$” can imply “($\exists \lambda \in {\mathbb {R}}$) ($\forall x\in {\mathbb {R}}^n$) $f(x) + \lambda g(x)\ge 0.$” Although the question has been answered by Xia et al. (Math Program 156:513–547, 2016), we propose a neat geometric proof for it (see Theorem 2): the ${\mathcal {S}}$-procedure holds when, and only when, the level set $\{g=0\}$ cannot separate the sublevel set $\{f<0\}.$ With such a separation property, we proceed to prove that, for two polynomials (g, f) both of degree 2, the image set of g over $\{f<0\}, g(\{f<0\})$, is always connected (see Theorem 4). It implies that the ${\mathcal {S}}$-procedure is a kind of the intermediate value theorem. As a consequence, we know not only the infimum of g over $\{f\le 0\}$, but the extended results when g over $\{f\le 0\}$ is unbounded from below or bounded but unattainable. The robustness and the sensitivity analysis of an optimization problem involving the pair (g, f) automatically follows. All the results in this paper are novel and fundamental in control theory and optimization.