  We propose the formulation of convex Generalized Disjunctive Programming
(GDP) problems using conic inequalities leading to conic GDP problems. We then
show the reformulation of conic GDPs into Mixed-Integer Conic Programming
(MICP) problems through both the big-M and hull reformulations. These
reformulations have the advantage that they are representable using the same
cones as the original conic GDP. In the case of the hull reformulation, they
require no approximation of the perspective function. Moreover, the MICP
problems derived can be solved by specialized conic solvers and offer a natural
extended formulation amenable to both conic and gradient-based solvers. We
present the closed form of several convex functions and their respective
perspectives in conic sets, allowing users to formulate their conic GDP
problems easily. We finally implement a large set of conic GDP examples and
solve them via the scalar nonlinear and conic mixed-integer reformulations.
These examples include applications from Process Systems Engineering, Machine
learning, and randomly generated instances. Our results show that the conic
structure can be exploited to solve these challenging MICP problems more
efficiently. Our main contribution is providing the reformulations, examples,
and computational results that support the claim that taking advantage of conic
formulations of convex GDP instead of their nonlinear algebraic descriptions
can lead to a more efficient solution to these problems.
