We address an optimal transmit power allocation problem that minimizes the outage probability of a secondary user (SU) who is allowed to coexist with a primary user (PU) in a narrowband spectrum sharing cognitive radio network, under a long term average transmit power constraint at the secondary transmitter (SU-TX) and an average interference power constraint at the primary receiver (PU-RX), with quantized channel state information (CSI) (including both the channels from SU-TX to SU-RX, denoted as g 1 and the channel from SU-TX to PU-RX, denoted as g 0 ) at the SU-TX. The optimal quantization regions in the vector channel space is shown to have a "stepwise" structure. With this structure, the above outage minimization problem can be explicitly formulated and solved by employing the Karush-Kuhn-Tucker (KKT) necessary optimality conditions to obtain a locally optimal quantized power codebook. A low-complexity near-optimal quantized power allocation algorithm is derived for the case of large number of feedback bits. More interestingly, we show that as the number of partition regions approaches infinity, the length of interval between any two adjacent quantization thresholds on the g 0 axis is asymptotically equal when the average interference power constraint is active. Similarly, we show that when the average interference power constraint is inactive, the ratio between any two adjacent quantization thresholds on the g 1 axis becomes asymptotically identical. Using these results, an explicit expression for the asymptotic SU outage probability at high rate quantization (as the number of feedback bits goes to infinity) is also provided, and is shown to approximate the optimal outage behavior extremely well for large number of bits of feedback via numerical simulations. Analysis on the extension to multiple secondary users case (cognitive multiple-access network) is also discussed. Numerical results illustrate that with only a few bits of feedback, the derived algorithms provide secondary outage performance very close to that with full CSI at the SU-TX.