In this paper, we are interested in finding the global minimum solution to a special class of non-convex quadratic programming problems with two quadratic inequality constraints, abbreviated as [Non-Cro \(/\) Sepa]. The class is defined geometrically so that the two constraint boundaries neither cross over, nor separate each other. We first formulate the geometrical idea of the feasible domain with set-inclusion relations. Then, the unsolvability of the two quadratic inequalities (when they have no common solution in \({\mathbb {R}}^n\) ) is studied with which we establish a new version of the \({\mathcal {S}}\) -procedure involving three quadratic functions. The \({\mathcal {S}}\) -procedure allows us to compute the optimal value via solving an SDP. Furthermore, with the same unsolvability result, we can either obtain an optimal solution or conclude the optimal value is indeed unattainable. As the scheme developed in the paper is very fundamental in mathematics, we expect that it can be generalized to solve other types of non-convex quadratically constrained quadratic programming.
