Extensions to generalized disjunctive programming: hierarchical structures and first-order logic

Optimization problems with discrete–continuous decisions are traditionally modeled in algebraic form via (non)linear mixed-integer programming. A more systematic approach to modeling such systems is to use generalized disjunctive programming (GDP), which extends the disjunctive programming paradigm proposed by Egon Balas to allow modeling systems from a logic-based level of abstraction that captures the fundamental rules governing such systems via algebraic constraints and logic. Although GDP provides a more general way of modeling systems, it warrants further generalization to encompass systems presenting a hierarchical structure. This work extends the GDP literature to address two major alternatives for modeling and solving systems with nested (hierarchical) disjunctions: explicit nested disjunctions and equivalent single-level disjunctions. We also provide theoretical proofs on the relaxation tightness of such alternatives, showing that explicitly modeling nested disjunctions is superior to the traditional approach discussed in literature for dealing with nested disjunctions.