## Abstract

Given a quadratic function f(x)=x^{T}Ax+2a^{T}x+a_{0} and its level sets, including {x∈R^{n}∣f(x)<0}, {x∈R^{n}∣f(x)≤0}, {x∈R^{n}∣f(x)=0}, {x∈R^{n}∣f(x)≥0}, {x∈R^{n}∣f(x)>0}, we derive conditions on the coefficients (A,a,a_{0}) 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∈R^{n}∣g(x)=x^{T}Bx+2b^{T}x+b_{0}=0} such that L^{-}⊂{x∈R^{n}∣g(x)<0} and L^{+}⊂{x∈R^{n}∣g(x)>0}. It has been shown that the particular case for {x∈R^{n}∣g(x)=0} to separate {x∈R^{n}∣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∈R^{n}∣g(x)=0} separates {x∈R^{n}∣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∈R^{n}∣g(x)=0} to separate {x∈R^{n}∣f(x)⋆0} where ⋆∈{<,≤,=,≥,>}, which is expected to provide a new tool in solving quadratic optimization problems.

Original language | English |
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Pages (from-to) | 803-829 |

Number of pages | 27 |

Journal | Journal of Global Optimization |

Volume | 88 |

Issue number | 4 |

DOIs | |

Publication status | Published - 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