An approach for minimizing a quadratically constrained fractional quadratic problem with application to the communications over wireless channels

Abstract

Studies for the cognitive model are relatively new in the literature; however there is a growing interest in the communication field nowadays. This paper considers the cognitive model in the communication field as the problem of minimizing a fractional quadratic problem, subject to two or more quadratic constraints in complex field. Although both denominator and numerator in the fractional problem are convex, this problem is not so simple since the quotient of convex functions is not convex in most cases. We first change the fractional problem into a non-fractional one. Second, we consider the semi-definite programming (SDP) method. For the problem with m (m≤2) constraints, we use the SDP relaxation and obtain the exact optimal solution. However, for the problem with m (m>2) constraints, we choose the randomization method to gain an approximation solution in the complex case. At last, we apply this method to practical communications over wireless channels with good results.

Keywords:

• fractional quadratic problem
• optimization
• SDP relaxation
• randomization method
• SNR

AMS Subject Classifications::

• 94A15
• 94A40
• 90C32
• 90C30

Acknowledgements

We are grateful to reviewers’ helpful comments and suggestions on revision of the paper. This research is supported by National Natural Science Foundation of China under grant numbers 10971017, 40974075 and the Knowledge Innovation Programs of Chinese Academy of Sciences KZCX2-YW-QN107.