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Diversity estimation of Pareto front (PF) approximations is a critical issue in the field of evolutionary multiobjective optimization. However, the existing diversity indicators are usually inappropriate for PF approximations with more than three objectives. Many of them can be utilized only when compared with approximations obtained by multiple multiobjective optimizers, which makes them difficult to use online. In this paper, we propose a unary diversity indicator based on reference vectors (DIR) to estimate the diversity of PF approximations for many-objective optimization. In DIR, a set of uniform and widespread reference vectors are generated. The coverage of each solution in the objective space is evaluated by the number of representative reference vectors it is associated with. The diversity (both spread and uniformity) is determined by the standard deviation of the coverage for all the solutions. The smaller value of DIR, the better the diversity of a PF approximation is. DIR can be applied to a unary approximation without any compared approximations needed. Thus, DIR is easy to use as either an offline indicator to estimate the performance of an optimizer or an online indicator for the selection of solutions in a MOEA. In the experimental studies, both the artificial and the real PF approximations generated by seven different many-objective algorithms are used to verify DIR as an offline indicator. The effects of the number of reference vectors on DIR are also investigated. In addition, as an online indicator, DIR is integrated into a Pareto-dominance-based evolutionary multiobjective optimizer, NSGA-II. The experimental studies show it has the significant performance enhancements over the original NSGA-II on many-objective optimization problems. [ABSTRACT FROM AUTHOR]

• Presenting an approach to dynamically adjusting the grouping of decision variables when the number of decision variables changes. • Providing a hybrid strategy of initializing subpopulations to rapidly respond to the change of an optimization problem. • Giving a distance-based strategy of selecting representatives from the other subpopulations when evaluating an individual. Multi-objective optimization problems (MOPs) with changing decision variables exist in the actual industrial production and daily life, which have changing Pareto sets and complex relations among decision variables and are difficult to solve. In this study, we present a cooperative co-evolutionary algorithm by dynamically grouping decision variables to effectively tackle MOPs with changing decision variables. In the presented algorithm, decision variables are grouped into a series of groups using maximum entropic epistasis (MEE) at first, with decision variables in different groups owning a weak dependency. Subsequently, a sub-population is generated to solve decision variables in each group with an existing multi-objective evolutionary algorithm (MOEA). Further, a complete solution including all the decision variables is achieved through the cooperation among sub-populations. Finally, when a decision variable is added or deleted from the existing problem, the grouping of decision variables is dynamically adjusted based on the correlation between the changed decision variable and existing groups. To verify the performance of the developed method, the presented method is compared with five popular methods by tackling eight benchmark optimization problems. The experimental results reveal that the presented method is superior in terms of diversity, convergence, and spread of solutions on most benchmark optimization problems. [ABSTRACT FROM AUTHOR]