Developing an efficient embedded feature selection method for both binary and multiclass classification problems is a fundamental topic to be further studied. In this paper, the conventional radial basis function (RBF) network is modified for joint feature selection and classification. By utilizing the anisotropic Gaussian basis function, an individual weight parameter is assigned to each feature for feature weighting. The L1-regularized loss function is designed by combining the softmax cross entropy loss with L1 regularization terms imposing on feature and output weights. A specialized active-set limited-memory projected Quasi-Newton (SAL-PQN) algorithm is proposed to minimize the L1-regularized loss function, which can optimize all the adjustable parameters and zero out the redundant feature and output weights simultaneously. Furthermore, a safe dynamic pruning strategy is integrated into the SAL-PQN algorithm for dynamic sparse training of the modified RBF network, which continuously excludes the dynamically zeroed-out feature weights from the training process. It is theoretically assured of yielding the same gradient updates as SAL-PQN, thus the relevant computation of the zeroed-out feature weights in the forward and backward propagations can be safely eliminated. The experimental results demonstrate the effectiveness and efficiency of the proposed methods for joint feature selection and classification, and illustrate the practical utility of the inherent feature weighting capability.