A QP1QC APPROACH FOR DECIDING WHETHER OR NOT TWO QUADRATIC SURFACES INTERSECT

Given two n-variate quadratic functions f(x)=x^TAx+2a^Tx+a₀and g(x)=x^TBx+2b^Tx+b₀we are interested in knowing whether or not the two hypersurfaces {xϵRⁿ:f(x)=0} and {xϵRⁿ:g(x)=0} intersect with each other. There are two ways of looking at this problem. In one respect, the famous Finsler–Calabi theorem (1936, 1964) asserts that if n ≥3 and f,g are quadratic forms, f=0 and g=0 has no common solution other than the trivial one, x=0, if and only if there exists a positive definite matrix pencil αA+β>0. The result is in general not true for nonhomogeneous quadratic functions. On the other hand, Levin (c. late 1970s) tried to directly solve the intersection curve of {xϵRⁿ:f(x)=0} and {xϵRⁿ:g(x)=0}, but it turned out to be way too ambitious. In this paper, we show that by incorporating the information about the unboundedness and the unattainability of several (at most 4) quadratic programming problems with one single quadratic constraint (QP1QC), the answer as to whether or not {xϵRⁿ:f(x)=0} and {xϵRⁿ:g(x)=0} intersect can be successfully determined.