Noritaka Tsukamoto

Evolutionary many-objective optimization: A short review

Whereas evolutionary multiobjective optimization (EMO) algorithms have successfully been used in a wide range of real-world application tasks, difficulties in their scalability to many-objective problems have also been reported. In this paper, first we demonstrate those difficulties through computational experiments. Then we review some approaches proposed in the literature for the scalability improvement of EMO algorithms. Finally we suggest future research directions in evolutionary many-objective optimization.

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).

Evolutionary many-objective optimization by NSGA-II and MOEA/D with large populations

Evolutionary multiobjective optimization (EMO) is an active research area in the field of evolutionary computation. EMO algorithms are designed to find a non-dominated solution set that approximates the entire Pareto front of a multiobjective optimization problem. Whereas EMO algorithms usually work well on two-objective and three-objective problems, their search ability is degraded by the increase in the number of objectives. One difficulty in the handling of many-objective problems is the exponential increase in the number of non-dominated solutions necessary for approximating the entire Pareto front. A simple countermeasure to this difficulty is to use large populations in EMO algorithms. In this paper, we examine the behavior of EMO algorithms with large populations (e.g., with 10,000 individuals) through computational experiments on multiobjective and many-objective knapsack problems with two, four, six, eight and ten objectives. We examine two totally different algorithms: NSGA-II and MOEA/D. NSGA-II is a Pareto dominance-based algorithm while MOEA/D uses scalarizing functions. Their search ability is examined for various specifications of the population size under the fixed computation load. That is, we use the total number of examined solutions as the stopping condition of each algorithm. Thus the use of a very large population leads to the termination at an early generation (e.g., 20th generation). It is demonstrated through computational experiments that the use of too large populations makes NSGA-II very slow and inefficient. On the other hand, MOEA/D works well even when it is executed with a very large population. We also discuss why MOEA/D works well even when the population size is unusually large.

Adaptation of Scalarizing Functions in MOEA/D: An Adaptive Scalarizing Function-Based Multiobjective Evolutionary Algorithm

It is well-known that multiobjective problems with many objectives are difficult for Pareto dominance-based algorithms such as NSGA-II and SPEA. This is because almost all individuals in a population are non-dominated with each other in the presence of many objectives. In such a population, the Pareto dominance relation can generate no strong selection pressure toward the Pareto front. This leads to poor search ability of Pareto dominance-based algorithms for many-objective problems. Recently it has been reported that better results can be obtained for many-objective problems by the use of scalarizing functions. The weighted sum usually works well in scalarizing function-based algorithms when the Pareto front is convex. However, we need other functions such as the weighted Tchebycheff when the Pareto front is non-convex. In this paper, we propose an idea of automatically choosing between the weighted sum and the weighted Tchebycheff for each individual in each generation. The characteristic feature of the proposed idea is to use the weighted Tchebycheff only when it is needed for individuals along non-convex regions of the Pareto front. The weighted sum is used for the other individuals in each generation. The proposed idea is combined with a high-performance scalarizing function-based algorithm called MOEA/D (multiobjective evolutionary algorithm based on decomposition) of Zhang and Li (2007). Effectiveness of the proposed idea is demonstrated through computational experiments on modified multiobjective knapsack problems with non-convex Pareto fronts.

Simultaneous use of different scalarizing functions in MOEA/D

The use of Pareto dominance for fitness evaluation has been the mainstream in evolutionary multiobjective optimization for the last two decades. Recently, it has been pointed out in some studies that Pareto dominance-based algorithms do not always work well on multiobjective problems with many objectives. Scalarizing function-based fitness evaluation is a promising alternative to Pareto dominance especially for the case of many objectives. A representative scalarizing function-based algorithm is MOEA/D (multiobjective evolutionary algorithm based on decomposition) of Zhang & Li (2007). Its high search ability has already been shown for various problems. One important implementation issue of MOEA/D is a choice of a scalarizing function because its search ability strongly depends on this choice. It is, however, not easy to choose an appropriate scalarizing function for each multiobjective problem. In this paper, we propose an idea of using different types of scalarizing functions simultaneously. For example, both the weighted Tchebycheff (Chebyshev) and the weighted sum are used for fitness evaluation. We examine two methods for implementing our idea. One is to use multiple grids of weight vectors and the other is to assign a different scalarizing function alternately to each weight vector in a single grid.

Effectiveness of scalability improvement attempts on the performance of NSGA-II for many-objective problems

Recently a number of approaches have been proposed to improve the scalability of evolutionary multiobjective optimization (EMO) algorithms to many-objective problems. In this paper, we examine the effectiveness of those approaches through computational experiments on multiobjective knapsack problems with two, four, six, and eight objectives. First we briefly review related studies on evolutionary many-objective optimization. Next we explain why Pareto dominance-based EMO algorithms do not work well on many-objective optimization problems. Then we explain various scalability improvement approaches. We examine their effects on the performance of NSGA-II through computational experiments. Experimental results clearly show that the diversity of solutions is decreased by most scalability improvement approaches while the convergence of solutions to the Pareto front is improved. Finally we conclude this paper by pointing out future research directions.

An empirical study on similarity-based mating for evolutionary multiobjective combinatorial optimization

We have already proposed a similarity-based mating scheme to recombine extreme and similar parents for evolutionary multiobjective optimization. In this paper, we examine the effect of the similarity-based mating scheme on the performance of evolutionary multiobjective optimization (EMO) algorithms. First we examine which is better between recombining similar or dissimilar parents. Next we examine the effect of biasing selection probabilities toward extreme solutions that are dissimilar from other solutions in each population. Then we examine the effect of dynamically changing the strength of this bias during the execution of EMO algorithms. Computational experiments are performed on a wide variety of test problems for multiobjective combinatorial optimization. Experimental results show that the performance of EMO algorithms can be improved by the similarity-based mating scheme for many test problems.

Many-objective test problems to visually examine the behavior of multiobjective evolution in a decision space

Many-objective optimization is a hot issue in the EMO (evolutionary multiobjective optimization) community. Since almost all solutions in the current population are non-dominated with each other in many-objective EMO algorithms, we may need a different fitness evaluation scheme from the case of two and three objectives. One difficulty in the design of many-objective EMO algorithms is that we cannot visually observe the behavior of multiobjective evolution in the objective space with four or more objectives. In this paper, we propose the use of many-objective test problems in a two- or three-dimensional decision space to visually examine the behavior of multiobjective evolution. Such a visual examination helps us to understand the characteristic features of EMO algorithms for many-objective optimization. Good understanding of existing EMO algorithms may facilitates their modification and the development of new EMO algorithms for many-objective optimization.

Indicator-based evolutionary algorithm with hypervolume approximation by achievement scalarizing functions

Pareto dominance-based algorithms have been the main stream in the field of evolutionary multiobjective optimization (EMO) for the last two decades. It is, however, well-known that Pareto-dominance-based algorithms do not always work well on many-objective problems with more than three objectives. Currently alternative frameworks are studied in the EMO community very actively. One promising framework is the use of an indicator function to find a good solution set of a multiobjective problem. EMO algorithms with this framework are called indicator-based evolutionary algorithms (IBEAs) where the hypervolume measure is frequently used as an indicator. IBEAs with the hypervolume measure have strong theoretical support and high search ability. One practical difficult of such an IBEA is that the hypervolume calculation needs long computation time especially when we have many objectives. In this paper, we propose an idea of using a scalarizing function-based hypervolume approximation method in IBEAs. We explain how the proposed idea can be implemented in IBEAs. We also demonstrate through computational experiments that the proposed idea can drastically decrease the computation time of IBEAs without severe performance deterioration.

Iterative approach to indicator-based multiobjective optimization

An emerging trend in the design of evolutionary multiobjective optimization algorithms is to directly optimize a quality indicator of non-dominated solution sets such as the hypervolume measure. Some algorithms have been proposed to search for a set of a pre-specified number of non-dominated solutions that maximizes the given quality indicator. In this paper, we propose an iterative approach to indicator-based evolutionary multiobjective optimization. The main feature of our approach is that only a single solution is obtained by its single run. Thus multiple runs are needed to find a solution set. In each run, our approach searches for a solution with the maximum contribution to the hypervolume of the solution set obtained by its previous runs. We discuss several issues related to the implementation of such an iterative approach.