Numerical Method for Generating the Entire Pareto Frontier in Multiobjective Optimization

In multidisciplinary optimization a designer solves a problem where there are different criteria usually contradicting each other. In general, the solution of such a problem is not unique. When seeking an optimal design, it is natural to exclude from the consideration any design solution which can be improved without deterioration of any discipline and violation of the constraints; in other words, a solution which can be improved without any trade-off. It leads to the Pareto optimal solutions. Mathematically, each Pareto point is a solution of the multidisciplinary optimization problem. Yet, it is important that the Pareto set is evenly distributed otherwise the representation of the Pareto surface may become inefficient. Although there are many numerical methods for vector optimization, only a few are able to generate the entire set of optimal solutions (Pareto frontier). It is very important for the decision maker, especially at the early design stages when most of the resources are committed, to be able to obtain maximum information on the Pareto surface at a minimal cost. The Physical Programming (PP) Method, suggested by Messac, is able to generate Pareto points on both convex and non-convex Pareto frontiers. The PP is substantially modified to make it simpler and more efficient for generating an even Pareto set. The algorithm does not provide non-Pareto solutions while local Pareto solutions may be easy recognized and removed in the framework of this method.