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Power control mechanisms play a critical role in underlay cognitive radio networks where primary users' quality of service should not be degraded by the secondary users transmission. Consequently, the inflicted interference power of the secondary transmitters on the primary receivers should be less than certain threshold levels that is required to be considered in designing power control algorithms for such networks. In this paper, we first propose a power reduction algorithm for the set of secondary transmitting links in order to guarantee that the members of this set do not violate the interference constraints set by primary users while maintaining their specific SINR quality of service...