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Decomposition-based evolutionary algorithms have become fairly popular formany-objective optimization in recent years. However, the existingdecomposition methods still are quite sensitive to the various shapes offrontiers of many-objective optimization problems (MaOPs). On the one hand, thecone decomposition methods such as the penalty-based boundary intersection(PBI) are incapable of acquiring uniform frontiers for MaOPs with very convexfrontiers. On the other hand, the parallel reference lines of the paralleldecomposition methods including the normal boundary intersection (NBI) mightresult in poor diversity because of under-sampling near the boundaries forMaOPs with concave frontiers. In this paper, a collaborative decompositionmethod is first proposed to integrate the advantages of parallel decompositionand cone decomposition to overcome their respective disadvantages. This methodinherits the NBI-style Tchebycheff function as a convergence measure toheighten the convergence and uniformity of distribution of the PBI method.Moreover, this method also adaptively tunes the extent of rotating an NBIreference line towards a PBI reference line for every subproblem to enhance thediversity of distribution of the NBI method. Furthermore, a collaborativedecomposition-based evolutionary algorithm (CoDEA) is presented formany-objective optimization. A collaborative decomposition-based environmentalselection mechanism is primarily designed in CoDEA to rank all the individualsassociated with the same PBI reference line in the boundary layer and pick outthe best ranks. CoDEA is compared with several popular algorithms on 85benchmark test instances. The experimental results show that CoDEA achieveshigh competitiveness benefiting from the collaborative decompositionmaintaining a good balance among the convergence, uniformity, and diversity ofdistribution.

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