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This paper presents how commonly used machine learning classifiers can be analyzed using a common framework of convex optimization. Four classifier models, the Support Vector Machine (SVM), the Least-Squares SVM (LSSVM), the Extreme Learning Machine (ELM), and the Margin Loss ELM (MLELM) are discussed to demonstrate how specific parametrizations of a general problem statement affect the classifier design and performance, and how ideas from the four different classifiers can be mixed and used together. Furthermore, 21 public domain benchmark datasets are used to experimentally evaluate five performance metrics of each model and corroborate the theoretical analysis. Comparison of classification accuracies under a nested cross-validation evaluation shows that with an exception all four models perform similarly on the evaluated datasets. However, the four classifiers command different amounts of computational resources for both testing and training. These requirements are directly linked to their formulations as different convex optimization problems.