On the convergence of decomposition algorithms in many-objective problems

Abstract

Decomposition methods have been considered for dealing with many-objective problems, since the Pareto-dominance selection was found to become ineffective as the number of objectives grow beyond four. As decomposition methods change the multiobjective problem into a set of single-objective problems, the difficulties found by evolutionary algorithms in many-objective optimization were expected to become alleviated. This paper studies the convergence properties of two decomposition schemes, respectively based on Euclidean norm and on Tchebyschev norm, in many-objective optimization. Numerical experiments show that the solution sequences obtained from Tchebyschev norm decomposition becomes stuck at a finite distance from the Pareto-set, while the sequences obtained from Euclidean norm decomposition may be adjusted such that an asymptotic convergence is achieved. Explanations for those different convergence behaviors are obtained from recently developed analytical tools.