In cognitive multiple access networks, primary-secondary feedback links are needed to convey secondary transmitter primary base station (STPB) channel gains from the primary base station (PBS) to the secondary base station (SBS). To reduce the amount of feedback exchange between PBS and SBS, this paper proposes a feedback control protocol called K-smallest channel gains (K-SCG) feedback protocol in which the PBS feeds back the KN smallest STPB channel gains, out of N of them, to the SBS. We study the performance of K-SCG feedback protocol for total power and interference limited (TPIL) networks when transmit powers of secondary users (SUs) are optimally allocated. In TPIL networks, transmit powers of SUs are limited by an average total power constraint as well as a constraint on the average total interference power that they cause to the PBS. It is shown that for KN = Nδ with δ ∈ (0, 1), K-SCG feedback protocol is asymptotically optimal, i.e., secondary network throughput under K-SCG and full feedback protocols scales according to 1/nh log log (N) where nh is a parameter obtained from the distribution of secondary transmitter secondary base station (STSB) channel power gains, and N is the number of SUs. It is also shown that for KN = o(N), the interference power at the PBS converges to zero almost surely and in mean as N becomes large. This result implies that for N large enough, the secondary network just requires the indices of SUs corresponding to the KN smallest STPB channel gains for performing jointly optimal user scheduling and power allocation rather than the actual realizations of STPB channel gains.