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.
