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This paper presents an adaptive downlink resource allocation strategy for multi-user orthogonal frequency division multiplexing (OFDM) cellular systems, which minimizes first the transmission bandwidth and then the transmission power. By analysis, it is shown that by keeping the minimum numbers of allocated sub-carriers to satisfy user rate requirements, the probability of interference occurrence in the sub-carriers is reduced and improves the expected link outage probability in the presence of flat fading and shadowing. The proposed bandwidth-constrained power minimization (BCPM) problem is formulated as a linear programming problem and used to derive three-step BCPM schemes. Simulation results in a cellular system to support voice and data services in presence of both frequency selective fading and shadowing show that the proposed BCPM schemes outperform the power minimization (PM) in terms of system outage and packet error rate.