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High system capacities can be achieved by controlling the transmitter power in multiuser radio systems. Power control with no constraint on the maximum power level has been studied extensively in earlier work. Transmitter power is at a premium in radio systems such as cellular systems and PCS. There is a limit on the maximum transmitter power especially at the terminals (e.g. mobile units and handsets) since the power comes from a battery. In this paper we study power control that maximizes the minimum carrier to interference ratio (CIR), with a constraint on the maximum power. The optimal power vector solution lies on the boundary of the constrained power vector set and achieves a balance in the CIR's. Results indicate that the constraints do not induce any stability problems. A distributed scheme with favourable convergence properties and close to optimum performance is presented. Simulation results …