In this paper, we address the constrained resource allocation problems arising in the context of spectrum sharing in cognitive radio networks utilizing a multi-dimensional formulation. Given the activity of the primary users (PUs), we consider multiple objectives and constraints, viz., sum rate, fairness, number of active secondary users (SUs), power consumption, and quality of service requirements (of both PUs and SUs). The three dimensions for the optimization task are the assignment of power, frequency, and antenna directionality to various SUs. Efficient heuristic algorithms are developed for five variations of the NP-hard optimization problems. Solution quality tradeoffs are shown for three algorithms, viz., convex relaxation with tree pruning, convex relaxation with gradual removal, and a genetic algorithm (GA); results show that the GA provides a reasonable balance between solution quality and computational effort. The multi-objective problems are solved using a modification of the NSGA-II evolutionary algorithm, obtaining a set of Paretooptimal solutions under computational constraints.