Geometric properties for level sets of quadratic functions

Huu-Quang Nguyen () and Ruey-Lin Sheu ()
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Huu-Quang Nguyen: Vinh University
Ruey-Lin Sheu: National Cheng Kung University

Journal of Global Optimization, 2019, vol. 73, issue 2, No 5, 349-369

Abstract: Abstract In this paper, we study some fundamental geometrical properties related to the $${\mathcal {S}}$$ S -procedure. Given a pair of quadratic functions (g, f), it asks when “ $$g(x)=0 \Longrightarrow ~ f(x)\ge 0$$ g ( x ) = 0 ⟹ f ( x ) ≥ 0 ” can imply “( $$\exists \lambda \in {\mathbb {R}}$$ ∃ λ ∈ R ) ( $$\forall x\in {\mathbb {R}}^n$$ ∀ x ∈ R n ) $$f(x) + \lambda g(x)\ge 0.$$ f ( x ) + λ g ( x ) ≥ 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 $${\mathcal {S}}$$ S -procedure holds when, and only when, the level set $$\{g=0\}$$ { g = 0 } cannot separate the sublevel set $$\{f

Keywords: $${\mathcal {S}}$$ S -procedure; Separation property; S-lemma with equality; Slater condition; Intermediate value theorem; Control theory (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10898-018-0706-2

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