A new method for estimating the relative positions of location-unaware nodes from the location-aware nodes and the received signal strength (RSS) between the nodes, in a wireless sensor network (WSN), is proposed. In the method, a regularization term is incorporated in the optimization problem leading to significant improvement in the estimation accuracy even in the presence of position errors of the location-aware nodes and distance errors between the nodes. The regularization term is appropriated weighted on the basis of the degree of connectivity between the nodes in the network. The method is formulated as a convex optimization problem using the semidefinite relaxation approach. Experimental comparisons with state-of-the-art competing methods show that the proposed method yields node positions that are much more accurate even in the presence of measurement errors.