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Decomposition-based multiobjective evolutionary algorithms (MOEAs) are a class of popular methods for solving multiobjective optimization problems (MOPs), and have been widely studied in numerical experiments and successfully applied in practice. However, we know little about these algorithms from the theoretical aspect. In this paper, we present a running time analysis of a simple MOEA with crossover based on the MOEA/D framework (MOEA/D-C) on four discrete optimization problems. Our rigorous theoretical analysis shows that the MOEA/D-C can obtain a set of Pareto optimal solutions to cover the Pareto front of these problems in expected running time apparently lower than the one without crossover. Moreover, the MOEA/D-C only needs to decompose an MOP into a few scalar optimization subproblems according to several simple weight vectors. This result suggests that the use of crossover in decomposition-based MOEA can simplify the setting of weight vector for different problems and make the algorithm more efficient. This study theoretically explains why some decomposition-based MOEAs work well in computational experiments and provides insights in design of MOEAs for MOPs in future research.

Somatic contiguous hypermutation (CHM) operators are important variation operators in artificial immune systems. The few existing theoretical studies are only concerned with understanding the optimization behavior of CHM operators on solving single-objective optimization problems. The MOEA/D framework is one of the most popular strategies for solving multi-objective optimization problems (MOPs). In this paper, we present a runtime analysis of using two CHM operators in MOEA/D framework for solving five benchmark MOPs, including four bi-objective and one many-objective problems. Our analyses show that the expected runtimes of CHM operators on the four bi-objective problems are better than or as good as that of the well-studied standard bit mutation operator. Moreover, using CHM operators in MOEA/D framework can improve the best known upper bound on the many-objective problem by a factor of n. This paper provides insight into understanding the optimization behavior of CHM operators in the well-known MOEA/D framework, and indicates that using the CHM operator in MOEA/D framework is a promising method for handling MOPs.

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