Separating disconnected quadratic level sets by other quadratic level sets

Huu Quang Nguyen, Ya Chi Chu, Ruey Lin Sheu

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Given a quadratic function f(x)=xTAx+2aTx+a0 and its level sets, including {x∈Rn∣f(x)<0}, {x∈Rn∣f(x)≤0}, {x∈Rn∣f(x)=0}, {x∈Rn∣f(x)≥0}, {x∈Rn∣f(x)>0}, we derive conditions on the coefficients (A,a,a0) for each of the five level sets to be disconnected with exactly two connected components, say L- and L+. Once so, we are interested in, when the two connected components can be separated by another quadratic level set {x∈Rn∣g(x)=xTBx+2bTx+b0=0} such that L-⊂{x∈Rn∣g(x)<0} and L+⊂{x∈Rn∣g(x)>0}. It has been shown that the particular case for {x∈Rn∣g(x)=0} to separate {x∈Rn∣f(x)<0} is equivalent to the S-lemma with equality, and thus answers the strong duality for min{f(x)∣g(x)=0}. In addition, the case when {x∈Rn∣g(x)=0} separates {x∈Rn∣f(x)=0} is closely related to the convexity of the joint range for a pair of quadratic functions (f, g). This paper completes the characterization for all five cases of separation, namely, for {x∈Rn∣g(x)=0} to separate {x∈Rn∣f(x)⋆0} where ⋆∈{<,≤,=,≥,>}, which is expected to provide a new tool in solving quadratic optimization problems.

Original languageEnglish
Pages (from-to)803-829
Number of pages27
JournalJournal of Global Optimization
Issue number4
Publication statusPublished - 2024 Apr

All Science Journal Classification (ASJC) codes

  • Business, Management and Accounting (miscellaneous)
  • Computer Science Applications
  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics


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