  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.
