Support vector machines (SVMs) are a supervised learning technique that can be applied in both binary and multi-class classification and regression tasks. SVMs seamlessly handle continuous and categorical variables. Their training is, however, both time- and memory-costly for large training data, and selecting an incorrect kernel function or its hyperparameters leads to suboptimal decision hyperplanes. In this paper, we introduce a memetic algorithm for evolving SVM training sets with adaptive radial basis function kernels to not only make the deployment of SVMs easier for emerging big data applications, but also to improve their generalization abilities over the unseen data. We build upon two observations: first, only a small subset of all training vectors, called the support vectors, contribute to the position of the decision boundary, hence the other vectors can be removed from the training set without deteriorating the performance of the model. Second, selecting different kernel hyperparameters for different training vectors may help better reflect the subtle characteristics of the space while determining the hyperplane. The experiments over almost 100 benchmark and synthetic sets showed that our algorithm delivers models outperforming both SVMs optimized using state-of-the-art evolutionary techniques, and other supervised learners.