In this paper, the problem of source localization in distributed multiple-input multiple-output (MIMO) radar using bistatic range measurements, which correspond to the sum of transmitter-to-target and target-to-receiver distances, is addressed. Our solution is based on the Lagrange programming neural network (LPNN), which is an analog neural computational technique for solving nonlinear constrained optimization problems according to the Lagrange multiplier theory. The local stability of the proposed positioning algorithm is also investigated. Furthermore, we have extended the LPNN based approach to more challenging scenarios, namely, when time synchronization among all antennas cannot be fulfilled, and there are position uncertainties in the MIMO radar transmit and receive elements. The optimality of the developed algorithms is demonstrated by comparing with the Cramér-Rao lower bound via computer simulations.