A Novel MINLP Reformulation for Nonlinear Generalized Disjunctive Programming (GDP) Problems

TL;DR

This paper introduces a new exact reformulation method for nonlinear generalized disjunctive programming (GDP) problems, enabling their conversion into standard MINLP problems without approximation or additional parameters, thus improving modeling and solution efficiency.

Contribution

The paper presents a novel reformulation technique for nonlinear GDP problems that preserves feasibility and convexity, avoiding the use of Big-M parameters and tolerances, and extends to complex logical constructs.

Findings

Exact reformulation of nonlinear GDP into MINLP

Preserves feasibility and convexity of constraints

Applicable to complex logical implications in optimization

Abstract

In optimization problems, often equations and inequalities are represented using if-else (implication) construct which is known to be equivalent to a disjunction. Such statements are modeled and incorporated in an optimization problem using Generalized Disjunctive Programming (GDP). GDP provides a systematic methodology to model optimization problems involving logic disjunctions, logic propositions, and algebraic equations. In order to take advantage of the existing MINLP solvers, GDP problems can be reformulated as the standard MINLP problems. In this work we propose a novel reformulation methodology for general GDP problems with nonlinear equality and inequality constraints. The proposed methodology provides an exact reformulation, maintains feasibility and convexity of the constraints, and, most importantly, does not require choosing a tolerance level and a Big-M parameter. We also demonstrate how the new reformulation approach can be used to convert the logic proposition represented using if-else (implication) construct into equations in the standard MINLP format. The conversion methodology is extended for variations of implication constructs that include implicit else blocks, sequential implication logic, multiple testing conditions, and nested implication blocks. The proposed approach is utilized to model physical and mechanical properties in a mathematical optimization tool that solves an MINLP problem to design commercial products.