Semidefinite relaxation and new conditions for sign-definiteness of the quadratic form under quadratic constraints

Use of the semidefinite relaxation in the problem of sign-definiteness of the quadratic form under quadratic constraints enables one to establish from the duality conditions an $S$-procedure. However, the $S$-procedure giving the necessary and sufficient conditions for signdefiniteness of the relaxed problem provides only the sufficient conditions for sign-definiteness for the original problem for the case of two and more quadratic constraints. This property is called the deficiency of $S$-procedure. A method was proposed enabling one in some cases to establish the conditional sign-definiteness in the case where the $S$-procedure provides a negative result. This method give the necessary and sufficient conditions for sign-definiteness in the two-dimensional case. An example was given.