Neural networks with non-linear rectified linear unit (ReLU) activation functions have demonstrated remarkable performance in many fields. It has been observed that a sufficiently wide and/or deep ReLU network can accurately fit the training data, with a small generalization error on the testing data. Nevertheless, existing analytical results on provably training ReLU networks are mostly limited to over-parameterized cases, or they require assumptions on the data distribution. In this paper, training a two-layer ReLU network for binary classification of linearly separable data is revisited. Adopting the hinge loss as classification criterion yields a non-convex objective function with infinite local minima and saddle points. Instead, a modified loss is proposed which enables (stochastic) gradient descent to attain a globally optimal solution. Enticingly, the solution found is globally optimal for the hinge loss too. In addition, an upper bound on the number of iterations required to find a global minimum is derived. To ensure generalization performance, a convex max-margin formulation for two-layer ReLU network classifiers is discussed. Connections between the sought max-margin ReLU network and the max-margin support vector machine are drawn. Finally, an algorithm-dependent theoretical quantification of the generalization performance is developed using classical compression bounds. Numerical tests using synthetic and real data validate the analytical results.