Radial basis function network-based autoregressive with exogenous input (RBF-ARX) models are useful in nonlinear system modelling and prediction. The identification of RBF-ARX models includes optimization of the (model lags, number of hidden nodes and state vector) and the parameters of the model. Previous works have usually ignored optimizations of the model's architecture. In this paper, the RBF-ARX architecture, which includes the selection of lags, number of nodes of the RBF network, lag orders and state vector, is encoded into a chromosome and is evolved simultaneously by a genetic algorithm (GA). This combines the advantages of the GA and the variable projection (VP) method to automatically generate a parsimonious RBF-ARX model with a high generalization performance. The highly efficient VP algorithm is used as a local search strategy to accelerate the convergence of the optimization. The experimental results demonstrate the effectiveness of the proposed method.