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Optimality conditions for Call Admission Control (CAC) problems with nonlinearly constrained feasibility regions and K classes of users are derived. The adopted model is a generalized stochastic knapsack, with exponentially distributed interarrival times of the objects. Call admission strategies are restricted to the family of Coordinate-Convex (CC) policies. For K = 2 classes of users, both general structural properties of the optimal CC policies and structural properties that depend on the revenue ratio are investigated. Then, the analysis is extended to the case K > 2. The theoretical results are exploited to narrow the set of admissible solutions to the associated knapsack problem, i.e., the set of CC policies to which an optimal one belongs. With respect to results available in the literature, less restrictive conditions on the optimality of the complete-sharing policy are obtained. To illustrate the role played by the theoretical results on the combinatorial CAC problem, simulation results are presented, which show how the number of candidate optimal CC policies dramatically decreases as the derived optimality conditions are imposed.