  We consider optimal state discrimination in a general convex operational
framework, so-called generalized probabilistic theories (GPTs), and present a
general method of optimal discrimination by applying the complementarity
problem from convex optimization. The method exploits the convex geometry of
states but not other detailed conditions or relations of states and effects. We
also show that properties in optimal quantum state discrimination are shared in
GPTs in general: i) no measurement sometimes gives optimal discrimination, and
ii) optimal measurement is not unique.
