Separating disconnected quadratic level sets by other quadratic level sets

Huu-Quang Nguyen (), Ya-Chi Chu () and Ruey-Lin Sheu ()
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Huu-Quang Nguyen: Vinh University
Ya-Chi Chu: Stanford University
Ruey-Lin Sheu: National Cheng Kung University

Journal of Global Optimization, 2024, vol. 88, issue 4, No 1, 803-829

Abstract: Abstract Given a quadratic function $$f(x) = x^T A x + 2 a^T x + a_0$$ f ( x ) = x T A x + 2 a T x + a 0 and its level sets, including $$\{x\in \mathbb {R}^n \mid f(x) 0\},$$ { x ∈ R n ∣ f ( x ) > 0 } , we derive conditions on the coefficients $$(A, a, a_0)$$ ( A , a , a 0 ) for each of the five level sets to be disconnected with exactly two connected components, say $$L^{-}$$ L - and $$L^{+}$$ L + . Once so, we are interested in, when the two connected components can be separated by another quadratic level set $$\{x\in \mathbb {R}^n \mid g(x) = x^T B x + 2b^T x + b_0 = 0\}$$ { x ∈ R n ∣ g ( x ) = x T B x + 2 b T x + b 0 = 0 } such that $$L^{-} \subset \{x\in \mathbb {R}^n \mid g(x) 0\}$$ L + ⊂ { x ∈ R n ∣ g ( x ) > 0 } . It has been shown that the particular case for $$\{x\in \mathbb {R}^n \mid g(x) = 0\}$$ { x ∈ R n ∣ g ( x ) = 0 } to separate $$\{x\in \mathbb {R}^n \mid f(x) \}$$ ⋆ ∈ { } , which is expected to provide a new tool in solving quadratic optimization problems.

Keywords: Non-convexity; Disconnectedness; Joint numerical range; Separating quadratic hypersurface; Hidden convexity; 90C20; 90C22; 90C26 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10898-023-01330-8

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