Semidefinite relaxation and new conditions of signdefinitness of quadratic forms under quadratic constraints

摘要

In this paper we consider the problem of the sign-definiteness of a quadratic form (QF) under quadratic constraints. The use of semidefinite relaxation allows us to derive an S-procedure from duality conditions. However, the S-procedure, which gives necessary and sufficient conditions for sign-definiteness for the relaxed problem, gives only sufficient conditions for sign-definiteness for the initial problem if number of quadratic constraints is two or more. In this paper the new approach is proposed, allowing for establishing of conditional sign definiteness in some cases, when the S-procedure doesn't give an answer. Results are illustrated by an example.