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Cooperative localization (also known as sensor network localization) using received signal strength (RSS) measurements when the source transmit powers are different and unknown is investigated. Previous studies were based on the assumption that the transmit powers of source nodes are the same and perfectly known which is not practical. In this paper, the source transmit powers are considered as nuisance parameters and estimated along with the source locations. The corresponding Cramér-Rao lower bound (CRLB) of the problem is derived. To find the maximum likelihood (ML) estimator, it is necessary to solve a nonlinear and nonconvex optimization problem, which is computationally complex. To avoid the difficulty in solving the ML estimator, we derive a novel semidefinite programming (SDP) relaxation technique by converting the ML minimization problem into a convex problem which can be solved efficiently. The algorithm requires only an estimate of the path loss exponent (PLE). We initially assume that perfect knowledge of the PLE is available, but we then examine the effect of imperfect knowledge of the PLE on the proposed SDP algorithm. The complexity analyses of the proposed algorithms are also studied in detail. Computer simulations showing the remarkable performance of the proposed SDP algorithm are presented.