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The rapid increase in wireless demand prompted the FCC to open up parts of the 5 GHz band for unlicensed access. This caught the interest of 4G/LTE providers, who wish to extend their LTE-A services to the unlicensed spectrum (LTE-U). In LTE-U, small-cell base stations aggregate unlicensed and licensed bands to increase the throughput. Wi-Fi/LTE-U coexistence is a challenging issue due to the different access mechanisms of these two systems, which may cause high collision rates and delays. By leveraging self-interference-suppression techniques, we propose joint mode/rate adaptation strategies for Wi-Fi/LTE-U coexistence. Specifically, a full-duplex enabled Wi-Fi station can transmit and receive data simultaneously (TR mode) to increase the throughput, or transmit and sense simultaneously (TS mode) to monitor the LTE-U activity. We model the LTE-U interference as a hidden Markov process, and solve the problem of jointly adapting Wi-Fi rates/modes using a framework of partially observable Markov decision process (POMDP). A detection approach based on the sliding window correlator is analyzed for the TS mode, which can differentiate between Wi-Fi and LTE-U signals. Our results indicate that our scheme provides 1.5 x (1.9 x) average throughput gain for Wi-Fi system in the low (high) SINR regime relative to a half-duplex-based scheme.