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Previous research efforts on OFDMA resource allocation have typically assumed the availability of perfect channel state information (CSI). Unfortunately, this is unrealistic, primarily due to channel estimation errors, and more importantly, channel feedback delay. In this paper, we develop optimal resource allocation algorithms for the OFDMA downlink assuming the availability of only partial (imperfect) CSI. We consider both continuous and discrete ergodic weighted sum rate maximization subject to total power constraints, and average bit error rate constraints for the discrete rate case. We approach these problems using a dual optimization framework, allowing us to solve these problems with O(MK) complexity per symbol for an OFDMA system with K used subcarriers and M active users, while achieving relative optimality gaps of less than 10-5 for continuous rates and less than 10-3 for discrete rates in simulations based on realistic parameters.