We develop a very general version of the hyperbola method which extends the
known method by Blomer and Br\"udern for products of projective spaces to a
very large class of toric varieties. We use it to count Campana points of
bounded log-anticanonical height on many split toric Q-varieties
with torus invariant boundary. We apply the strong duality principle in linear
programming to show the compatibility of our results with the conjectured
asymptotic.Comment: 47 pages, Section 4.1 has been added and the main results have been
strengthene