Yusuke Nojima

Evolutionary many-objective optimization

In this paper, we first explain why many-objective problems are difficult for Pareto dominance-based evolutionary multiobjective optimization algorithms such as NSGA-II and SPEA. Then we explain recent proposals for the handling of many-objective problems by evolutionary algorithms. Some proposals are examined through computational experiments on multiobjective knapsack problems with two, four and six objectives. Finally we discuss the viability of many-objective genetic fuzzy systems (i.e., the use of many-objective genetic algorithms for the design of fuzzy rule-based systems).

Performance of Decomposition-Based Many-Objective Algorithms Strongly Depends on Pareto Front Shapes

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

Modified Distance Calculation in Generational Distance and Inverted Generational Distance

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Difficulties in specifying reference points to calculate the inverted generational distance for many-objective optimization problems

-dominance based evolutionary algorithm (

Pareto Fronts of Many-Objective Degenerate Test Problems

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A Study on Performance Evaluation Ability of a Modified Inverted Generational Distance Indicator

-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.

Review of coevolutionary developments of evolutionary multi-objective and many-objective algorithms and test problems

In this paper, we propose the use of modified distance calculation in generational distance (GD) and inverted generational distance (IGD). These performance indicators evaluate the quality of an obtained solution set in comparison with a pre-specified reference point set. Both indicators are based on the distance between a solution and a reference point. The Euclidean distance in an objective space is usually used for distance calculation. Our idea is to take into account the dominance relation between a solution and a reference point when we calculate their distance. If a solution is dominated by a reference point, the Euclidean distance is used for their distance calculation with no modification. However, if they are non-dominated with each other, we calculate the minimum distance from the reference point to the dominated region by the solution. This distance can be viewed as an amount of the inferiority of the solution (i.e., the insufficiency of its objective values) in comparison with the reference point. We demonstrate using simple examples that some Pareto non-compliant results of GD and IGD are resolved by the modified distance calculation. We also show that IGD with the modified distance calculation is weakly Pareto compliant whereas the original IGD is Pareto non-compliant.

Comparing solution sets of different size in evolutionary many-objective optimization

Recently the inverted generational distance (IGD) measure has been frequently used for performance evaluation of evolutionary multi-objective optimization (EMO) algorithms on many-objective problems. When the IGD measure is used to evaluate an obtained solution set of a many-objective problem, we have to specify a set of reference points as an approximation of the Pareto front. The IGD measure is calculated as the average distance from each reference point to the nearest solution in the solution set, which can be viewed as an approximate distance from the Pareto front to the solution set in the objective space. Thus the IGD-based performance evaluation totally depends on the specification of reference points. In this paper, we illustrate difficulties in specifying reference points. First we discuss the number of reference points required to approximate the entire Pareto front of a many-objective problem. Next we show some simple examples where the uniform sampling of reference points on the known Pareto front leads to counter-intuitive results. Then we discuss how to specify reference points when the Pareto front is unknown. In this case, a set of reference points is usually constructed from obtained solutions by EMO algorithms to be evaluated. We show that the selection of EMO algorithms used to construct reference points has a large effect on the evaluated performance of each algorithm.

Characteristics of many-objective test problems and penalty parameter specification in MOEA/D

In general, an M-objective continuous optimization problem has an (M - 1)-dimensional Pareto front in the objective space. If its dimension is smaller than (M - 1), it is called a degenerate Pareto front. Deb-Thiele-Laumanns-Zitzler (DTLZ)5 and Walking Fish Group (WFG)3 have often been used as many-objective continuous test problems with degenerate Pareto fronts. However, it was noted that DTLZ5 has a nondegenerate part of the Pareto front. Constraints have been proposed to remove the nondegenerate part. In this letter, first we show that WFG3 also has a nondegenerate part. Then, we derive constraints to remove the nondegenerate part. Finally, we show that the existence of the nondegenerate part makes WFG3 an interesting test problem through computational experiments.

Performance comparison of NSGA-II and NSGA-III on various many-objective test problems

The inverted generational distance (IGD) has been frequently used as a performance indicator for many-objective problems where the use of the hypervolume is difficult. However, since IGD is not Pareto compliant, it is possible that misleading Pareto incompliant results are obtained. Recently, a simple modification of IGD was proposed by taking into account the Pareto dominance relation between a solution and a reference point when their distance is calculated. It was also shown that the modified indicator called IGD+ is weakly Pareto compliant. However, actual effects of the modification on performance comparison have not been examined. Moreover, IGD+ has not been compared with other distance-based weakly Pareto compliant indicators such as the additive epsilon indicator and the D1 indicator (i.e., IGD with the weighted achievement scalarizing function). In this paper, we examine the effect of the modification by comparing IGD+ with IGD for multiobjective and many-objective problems. In computational experiments, we generate a large number of ordered pairs of non-dominated solution sets where one is better than the other. Two solution sets in each pair are compared by the above-mentioned performance indicators. We examine whether each indicator can correctly say which solution set is better between them.