This paper proposes an adaptive sparsity-based direct position determination (DPD) appoach to locate multiple targets in the case of time-varying channels. The novel feature of this method is to dynamically adjust both the overcomplete basis and the sparse solution based on a two-step dictionary learning (DL) framework. The method first performs supervised offline DL by using the quadratic programming approach, and then the dictionary is continuously updated in an incremental fashion to adapt to the time-varying channel during the online stage. Furthermore, the method does not need the number of emitters a prior. Simulation results demonstrate the performance of the proposed algorithm on the location estimation accuracy.