We present a new method for the incremental training of multiclass Support Vector Machines that provides computational efficiency for training problems in the case where the training data collection is sequentially enriched and dynamic adaptation of the classifier is required. An auxiliary function that incorporates some desired characteristics in order to provide an upper bound of the objective function which summarizes the multiclass classification task has been designed and the global minimizer for the enriched dataset is found using a warm start algorithm, since faster convergence is expected when starting from the previous global minimum. Experimental evidence on two data collections verified that our method is faster than retraining the classifier from scratch, while the achieved classification accuracy is maintained at the same level.