A collaborative decomposition-based evolutionary algorithm integrating normal and penalty-based boundary intersection methods for many-objective optimization

Decomposition-based evolutionary algorithms have recently become fairly popular for many-objective optimization owing to their excellent selection pressure. However, existing decomposition methods are still quite sensitive to the various shapes of the frontiers of many-objective optimization problems (MaOPs). On the one hand, the penalty-based boundary intersection (PBI) method is incapable of acquiring uniform frontiers for MaOPs with very convex frontiers due to the radial distribution of the reference lines. On the other hand, the parallel reference lines of the normal boundary intersection (NBI) method often result in poor diversity for MaOPs with concave frontiers because of under-sampling near the boundaries. In this paper, a collaborative decomposition (CoD) method is first proposed to integrate the advantages of the PBI and NBI methods to overcome their respective disadvantages. This method inherits the NBI-style Tchebycheff function as a convergence measure to improve the convergence and uniformity of the distribution of the PBI method. Moreover, this method also adaptively tunes the extent to which an NBI reference line is rotated towards a PBI reference line for every boundary subproblem to enhance the diversity of the distribution of the NBI method. Furthermore, a CoD-based evolutionary algorithm (CoDEA) is presented for many-objective optimization. A CoD-based ranking strategy is primarily designed in the CoDEA to rank all the individuals associated with every boundary subproblem according to the CoD aggregation function and determine the best ranks. The proposed algorithm is compared with several popular many-objective evolutionary algorithms on 85 benchmark test instances. The experimental results show that the CoDEA achieves high competitiveness, benefiting from the CoD method.