Generalized Neyman-Pearson lemma via convex duality

We extend the classical Neyman-Pearson theory for testing composite hypothe ses versus composite alternatives, using a convex duality approach, first e mployed by Witting. Results of Aubin and Ekeland from non-smooth convex ana lysis are used along with a theorem of Komlos, in order to establish the ex istence of a max-min optimal test in considerable generality, and to invest igate its properties. The theory is illustrated on representative examples involving Gaussian measures on Euclidean and Wiener space.