Detecting local convexity on the Pareto surface

Recent regain in interest in multi-criteria optimization approaches to provide a designer with multiple solutions to select from and support decisions has led to various methodologies to generate such solutions and possibly approximate the Pareto set. This paper introduces the notions of H- and w-convexities, and develops a simple method to identify local convexity of Pareto hyper-surfaces since their shape can dictate the choice of the method to use to obtain Pareto solutions and possibly to build an approximation of that set. The method is based on comparing the results from the weighting method to those of the Tchebycheff method at any point on the Pareto hyper-surface. If, under some conditions, the points obtained from the two methods are identical or not, a local convexity or its lack can be assumed at that location and in its immediate neighbourhood. A numerical example is included.