🤖 AI Summary
This work addresses the critical limitation of existing multi-objective optimization methods—such as weighted sum, NNC, and NBI—which frequently fail to fully capture the Pareto front in problems with three or more objectives, often exhibiting omission rates exceeding 50%. To overcome this, the authors propose the Generalized Normal Constraint (GNC) method, which establishes a unified geometric and computational framework. By integrating Pareto front mesh construction with a normal constraint mechanism, GNC structurally guarantees 100% coverage of the feasible Pareto region for any n-objective problem. This approach fundamentally resolves the factorial degradation in coverage that plagues conventional techniques as the number of objectives increases. Theoretically ensuring solution set completeness, GNC significantly outperforms current methods and offers a more comprehensive solution for multi-objective optimization in fields such as engineering design and economic decision-making.
📝 Abstract
This paper presents a comprehensive geometric and computational framework for the generation of the complete Pareto frontier. Several existing methods are structurally unable to capture the complete admissible Pareto region. These include widely used methods such as the weighted sum, compromise programming, 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 this paper is structurally capable of capturing 100% of the admissible Pareto region.
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 insightful examples. Multiobjective optimization plays an important role in a broad range of applications, including economics, product design, and engineering management. Accordingly, the ability of an optimization method to generate a representative subset spanning the complete Pareto frontier is of fundamental importance.