In this paper, we investigate the tradeoff between increasing the secondary users' (SUs) transmission rate and reducing the interference levels at the primary users (PUs) for orthogonal-frequency-division-multiplexing-based cognitive radio systems. To achieve this target, we formulate a generalized multiobjective optimization (MOOP) problem that jointly maximizes the transmission rate of the SU and minimizes the cochannel interference (CCI) and adjacent channel interference (ACI) to existing PUs. We additionally constrain the allowed CCI and ACI to the PUs to guarantee the PUs' protection from harmful interference. The MOOP problem is solved by linearly combining the normalized competing objective functions - through weighting coefficients - into a single objective function. Prior work in the literature that maximizes the SU transmission rate can be considered as a special case of the generalized MOOP problem by setting the weighting coefficients associated with interference minimization to zero. Since estimating the full channel state information (CSI) of the links between the SU transmitter and the PU receivers is practically challenging, we assume only partial CSI knowledge of these links. Simulation results illustrate the performance of the proposed algorithm and quantify the SU performance loss due to incomplete CSI knowledge. Furthermore, the proposed algorithm is compared with state-of-the-art techniques, and our performance results show that the proposed algorithm is more energy aware, yet with reduced complexity.