Higher-Order Dimensionality

Abstract

While in two dimensions the PF is always a curve, in three dimensions it can be either a curve or a surface, depending on the structure of the objective space. In multiple dimensions ${n_f} \geq 3$, representing the PF is still an open problem. However, in most practical cases, a reasonable simplification consists of considering the hortogonal projections of the n_f-dimensional objective space onto planes spanned by all possible pairs of objectives (f_i,f_j), $i,j = 1,{n_f},\,\,\,i \ne j$. Successively, given a pair (i,j), the relevant front can be identified, irrespective of all the other objectives f_k, $k = 1,{n_f}$, $k \ne i$, $k \ne j$. To clarify the technique, an application is here presented.