Aggregate Objective Functions and Pareto Frontiers: Required Relationships and Practical Implications

This paper addresses the problem of capturing Pareto optimal points on non-convex Pareto frontiers, which are encountered in nonlinear multiobjective optimization problems in computational engineering design optimization. The emphasis is on the choice of the aggregate objective function (AOF) of the objectives that is employed to capture Pareto optimal points. A fundamental property of the aggregate objective function, the admissibility property, is developed and its equivalence to the coordinatewise increasing property is established. Necessary and sufficient conditions for such an admissible aggregate objective function to capture Pareto optimal points are derived. Numerical examples illustrate these conditions in the biobjective case. This paper demonstrates in general terms the limitation of the popular weighted-sum AOF approach, which captures only convex Pareto frontiers, and helps us understand why some commonly used AOFs cannot capture desirable Pareto optimal points, and how to avoid this situation in practice. Since nearly all applications of optimization in engineering design involve the formation of AOFs, this paper is of direct theoretical and practical usefulness.