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This paper presents joint power allocation and interference mitigation techniques for the downlink of CDMA systems which employ multiple relays and the amplify and forward cooperation strategy. We propose a joint constrained optimization framework that considers the allocation of power levels across the relays subject to individual power constraints and the design of linear receivers for interference suppression. We derive constrained minimum mean-squared error (MMSE) expressions for the parameter vectors that determine the optimal power levels across the relays and the linear receivers. In order to solve the proposed optimization problem efficiently, we develop a joint adaptive power allocation and interference suppression stochastic gradient (SG) algorithm. The proposed SG algorithm mitigates the interference by adjusting the power levels across the relays and estimating the parameters of the linear receiver. An SG channel estimation algorithm is also derived to determine the coefficients of the channels across the base station, the relays and the destination terminal. The results of simulations show that the proposed methods obtain significant gains in performance and capacity over non-cooperative systems and cooperative schemes with equal power allocation.