Extensions to Generalized Disjunctive Programming: Hierarchical Structures and First-order Logic

TL;DR

This paper extends Generalized Disjunctive Programming (GDP) to better model hierarchical systems with nested disjunctions, providing theoretical insights and demonstrating the advantages of explicit nested disjunctions over flattening methods.

Contribution

It introduces hierarchical structures into GDP, analyzes different modeling approaches for nested disjunctions, and proves the relaxation tightness of explicit nested disjunctions.

Findings

Explicit nested disjunctions offer tighter relaxations.

Flattening nested disjunctions can lead to looser relaxations.

Theoretical proofs support the superiority of hierarchical modeling.

Abstract

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 three major alternatives for modeling and solving systems with nested (hierarchical) disjunctions: explicit nested disjunctions, equivalent single-level disjunctions, and flattening via basic steps. 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.