Accurate nonlinear dynamic models of process operations such as start-ups and shut-downs include state-dependent events that trigger discrete changes to the describing equations, and are best analyzed within a hybrid systems framework. The automated design of an optimal process operation can thus be formulated as a dynamic optimization problem with a hybrid system embedded. This paper describes recent progress on the development of suitable deterministic algorithms for the global solution of these problems. A method for constructing convex relaxations of general nonconvex nonlinear programming (NLP) problems with linear dynamic systems embedded is presented. These convex relaxations are then extended to multistage problems with model switches between the stages. Finally, integer variables are introduced to represent alternative sequences of model switches. The ability to construct convex relaxations enables existing nonconvex mixed nonlinear programming (MINLP) algorithms to be applied to find the global solution of the resulting mixed integer dynamic optimization (MIDO) problems.