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

Convex Mixed-Integer Nonlinear Programs Derived from Generalized Disjunctive Programming using Cones