Objective Dimension and Problem Structurein Multiobjective Optimization Problems

Abstract

Multiobjective optimization seeks simultaneous minimization of multiple scalar functions on $\mathbb {R}^n$. Unless weighted sums are made to replace the vector functions arising thus, such an optimization requires some partial- or quasi-ordering of points in the search space based on comparisons between the values attained by the functions to be optimized at those points. Many such orders can be defined, and search-based (mainly heuristic) optimization algorithms make use of such orders implicitly or explicitly for refining and accelerating search. In this work, such relations are studied by modeling them as graphs. Information apparent in the structure of such graphs is studied in the form of degree distribution. It is found that when the objective dimension grows, the degree distribution tends to follow a power-law. This can be a new beginning in the study of escalation of hardness of problems with dimension, as also a basis for designing new heuristics.