Geometric properties for level sets of quadratic functions

• Published in: Journal of global optimization Vol. 73; no. 2; pp. 349 - 369
• Main Authors: Nguyen, Huu-Quang, Sheu, Ruey-Lin
• Format: Journal Article
• Language: English
• Published: New York Springer US 15.02.2019; Springer; Springer Nature B.V
• Subjects: Computer Science; Control theory; Mathematical analysis; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Real Functions; Intermediate value theorem; procedure; Control theory; S-lemma with equality; Slater condition; Separation property
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-018-0706-2

In this paper, we study some fundamental geometrical properties related to the S -procedure. Given a pair of quadratic functions ( g ,  f ), it asks when “ g ( x ) = 0 ⟹ f ( x ) ≥ 0 ” can imply “( ∃ λ ∈ R ) ( ∀ x ∈ R n )  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 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 S -procedure is a kind of the intermediate value theorem. As a consequence, we know not only the infimum of g over { f ≤ 0 } , but the extended results when g over { f ≤ 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.