Recently, a number of high performance many-objective evolutionary algorithms with systematically generated weight vectors have been proposed in the literature. Those algorithms often show surprisingly good performance on widely used DTLZ and WFG test problems. The performance of those algorithms has continued to be improved. The aim of this paper is to show our concern that such a performance improvement race may lead to the overspecialization of developed algorithms for the frequently used many-objective test problems. In this paper, we first explain the DTLZ and WFG test problems. Next, we explain many-objective evolutionary algorithms characterized by the use of systematically generated weight vectors. Then we discuss the relation between the features of the test problems and the search mechanisms of weight vector-based algorithms such as multiobjective evolutionary algorithm based on decomposition (MOEA/D), nondominated sorting genetic algorithm III (NSGA-III), MOEA/dominance and decomposition (MOEA/DD), and θ-dominance based evolutionary algorithm (θ-DEA). Through computational experiments, we demonstrate that a slight change in the problem formulations of DTLZ and WFG deteriorates the performance of those algorithms. After explaining the reason for the performance deterioration, we discuss the necessity of more general test problems and more flexible algorithms.

In this paper, we propose a new method for solving multiobjective optimization problems with a large number of decision variables. The proposed method called weighted optimization framework is intended to serve as a generic method that can be used with any population-based metaheuristic algorithm. After explaining some general issues of large-scale optimization, we introduce a problem transformation scheme that is used to reduce the dimensionality of the search space and search for improved solutions in the reduced subspace. This involves so-called weights that are applied to alter the decision variables and are also subject to optimization. Our method relies on grouping mechanisms and employs a population-based algorithm as an optimizer for both original variables and weight variables. Different grouping mechanisms and transformation functions within the framework are explained and their advantages and disadvantages are examined. Our experiments use test problems with 2-3 objectives 40-5000 variables. Using our approach on three well-known algorithms and comparing its performance with other large-scale optimizers, we show that our method can significantly outperform most existing methods in terms of solution quality as well as convergence rate on almost all tested problems for many-variable instances.

The hypervolume indicator has frequently been used for comparing evolutionary multi-objective optimization (EMO) algorithms. A reference point is needed for hypervolume calculation. However, its specification has not been discussed in detail from a viewpoint of fair performance comparison. A slightly worse point than the nadir point is usually used for hypervolume calculation in the EMO community. In this paper, we propose a reference point specification method for fair performance comparison of EMO algorithms. First, we discuss the relation between the reference point specification and the optimal distribution of solutions for hypervolume maximization. It is demonstrated that the optimal distribution of solutions strongly depends on the location of the reference point when a multi-objective problem has an inverted triangular Pareto front. Next, we propose a reference point specification method based on theoretical discussions on the optimal distribution of solutions. The basic idea is to specify the reference point so that a set of well-distributed solutions over the entire linear Pareto front has a large hypervolume and all solutions in such a solution set have similar hypervolume contributions. Then, we examine whether the proposed method can appropriately specify the reference point through computational experiments on various test problems. Finally, we examine the usefulness of the proposed method in a hypervolume-based EMO algorithm. Our discussions and experimental results clearly show that a slightly worse point than the nadir point is not always appropriate for performance comparison of EMO algorithms.

The hypervolume and the inverted generational distance (IGD) have been frequently used for the comparison of evolutionary multiobjective optimization (EMO) algorithms. For the hypervolume, the relation between the location of a reference point and the optimal distribution of solutions has been studied in the literature. However, such a relation has not been studied for the IGD whereas IGD-based comparison results depend on the specification of reference points. Our intention is to clearly demonstrate the dependency of IGD-based comparison results on reference point specification. First we explain difficulties of fair comparison in the following two cases: One is the use of all non-dominated solutions among obtained solutions by compared algorithms as reference points, and the other is the use of a small number of uniformly sampled reference points. Discussions on these two cases show the necessity of a large number of uniformly sampled reference points on the entire Pareto front. Then, we show a bias of the IGD with such a reference point set through computational experiments. It is shown that the IGD tends to favor a solution set with much smaller diversity than a fully expanded solution set over the entire Pareto front. Finally, we propose a new specification method where reference points are uniformly sampled not only from the Pareto front but also from outside the Pareto front.