Generalized Normal Constraint (GNC): A Complete Geometric Generalization of the NNC Method

Achille Messac, Blayne Montaque

This paper presents a unified geometric, mathematical, and computational framework for the generation of the $complete$ admissible Pareto frontier. Several existing methods are structurally unable to capture the complete admissible Pareto frontier. These include widely used methods such as the weighted sum, the Normal Boundary Intersection (NBI) method, and the Normalized Normal Constraint (NNC) method. NNC and NBI, which share the same Pareto-generation grid construction, are structurally unable to capture 50% of the admissible Pareto region for tri-objective problems. More generally, for an $n$-objective problem, the admissible capture fraction decreases factorially as $1/(n-1)!$, and the corresponding missed fraction increases to $1-1/(n-1)!$. By contrast, the newly developed Generalized Normal Constraint (GNC) method introduced in the present work is structurally capable of capturing the complete admissible Pareto frontier. The proposed GNC method is formulated for general $n$-objective optimization problems and is developed through a unified geometric, mathematical, and computational framework supported by computational examples. Multiobjective optimization plays an important role in a broad range of applications, including economics, product design, and engineering management. Accordingly, the ability of a Pareto-generation method to generate a representative subset spanning the $complete$ admissible Pareto frontier is of fundamental importance for multiobjective optimization.