Semidefinite Relaxation and New Conditions for Sign-Definiteness of the Quadratic Form under Quadratic Constraints


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Abstract

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.

About the authors

L. B. Rapoport

Trapeznikov Institute of Control Sciences

Author for correspondence.
Email: LBRapoport@gmail.com
Russian Federation, Moscow

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