Hisao Ishibuchi

On the effect of reference point in MOEA/D for multi-objective optimization

Abstract Multi-objective evolutionary algorithm based on decomposition (MOEA/D) has continuously proven effective for multi-objective optimization. So far, the effect of weight vectors and scalarizing methods in MOEA/D has been intensively studied. However, the reference point which serves as the starting point of reference lines (determined by weight vectors) is yet to be well studied. This study aims to fill in this research gap. Ideally, the ideal point of a multi-objective problem could serve as the reference point, however, since the ideal point is often unknown beforehand, the reference point has to be estimated (or specified). In this study, the effect of the reference point specified in three representative manners, i.e., pessimistic, optimistic and dynamic (from optimistic to pessimistic), is examined on three sets of benchmark problems. Each set of the problems has different degrees of difficulty in convergence and spread. Experimental results show that (i) the reference point implicitly impacts the convergence and spread performance of MOEA/D; (ii) the pessimistic specification emphasizes more of exploiting existing regions and the optimistic specification emphasizes more of exploring new regions; (iii) the dynamic specification can strike a good balance between exploitation and exploration, exhibiting good performance for most of the test problems, and thus, is commended to use for new problems.

Pattern Classification with Linguistic Rules

Linguistic rules are fuzzy rules described by linguistic terms such as small and large. Here we discuss pattern classification with linguistic rules. The main advantage of using linguistic rules is their high interpretability. We can construct linguistically interpretable fuzzy rule-based classification systems using linguistic rules. First we briefly explain fuzzy rules for function approximation. Next we explain fuzzy rules and fuzzy reasoning for pattern classification. Then we explain linguistic rule extraction from numerical data. Finally we show some future research topics on pattern classification with linguistic rules.

Reference point specification in hypervolume calculation for fair comparison and efficient search

Hypervolume has been frequently used as a performance indicator for comparing evolutionary multiobjective optimization (EMO) algorithms. Hypervolume has been also used in indicator-based algorithms. Whereas a reference point is needed for hypervolume calculation, its specification has not been discussed in detail from a viewpoint of fair comparison. This may be because a slightly worse reference point than the nadir point seems to work well. In this paper, we tackle this issue: How to specify a reference point for fair comparison. First we discuss an appropriate specification of a reference point for multiobjective problems. Our discussions are based on the well-known theoretical results about the optimal solution distribution for hypervolume maximization. Next we examine various specifications by computational experiments. Experimental results show that a slightly worse reference point than the nadir point works well only for test problems with triangular Pareto fronts. Then we explain why this specification is not always appropriate for test problems with inverted triangular Pareto fronts. We also report a number of solution sets obtained by SMS-EMOA with various specifications of a reference point.

Benchmarking Multi- and Many-Objective Evolutionary Algorithms Under Two Optimization Scenarios

Recently, a large number of multi-objective evolutionary algorithms (MOEAs) for many-objective optimization problems have been proposed in the evolutionary computation community. However, an exhaustive benchmarking study has never been performed. As a result, the performance of the MOEAs has not been well understood yet. Moreover, in almost all previous studies, the performance of the MOEAs was evaluated based on nondominated solutions in the final population at the end of the search. Such traditional benchmarking methodology has several critical issues. In this paper, we exhaustively investigate the anytime performance of 21 MOEAs using an unbounded external archive (UEA), which stores all nondominated solutions found during the search process. Each MOEA is evaluated under two optimization scenarios called UEA and reduced UEA in addition to the standard final population scenario. These two scenarios are more practical in real-world applications than the final population scenario. Experimental results obtained under the two scenarios are significantly different from the previously reported results under the final population scenario. For example, results on the Walking Fish Group test problems with up to six objectives indicate that some recently proposed MOEAs are outperformed by some classical MOEAs. We also analyze the reason why some classical MOEAs work well under the UEA and the reduced UEA scenarios.

A Framework for Large-Scale Multiobjective Optimization Based on Problem Transformation

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.

Deep Takagi–Sugeno–Kang Fuzzy Classifier With Shared Linguistic Fuzzy Rules

In many practical applications of classifiers, not only high accuracy but also high interpretability is required. Among a wide variety of existing classifiers, Takagi–Sugeno–Kang (TSK) fuzzy classifiers may be one of the best choices for achieving a good balance between interpretability and accuracy. In order to further improve their accuracy without losing their interpretability, we propose a highly interpretable deep TSK fuzzy classifier HID-TSK-FC (deep shared-linguistic-rule-based TSK fuzzy classifier) based on the concept of shared linguistic fuzzy rules. The proposed classifier has two characteristics: One is a stacked hierarchical structure of component TSK fuzzy classifiers for high accuracy, and the other is the use of interpretable linguistic rules with the same set of linguistic labels for all inputs. High interpretability is achieved at each layer by using the same set of linguistic values for all inputs, including the outputs from the previous layers in the stacked hierarchical structure. We show that a linguistic rule with the outputs from the previous layers as its inputs is equivalent to a fuzzy rule with a nonlinear consequent or a linear consequent with a certainty factor. We also show that HID-TSK-FC is mathematically equivalent to a novel TSK fuzzy classifier with shared interpretable linguistic fuzzy rules. Promising performance of HID-TSK-FC is demonstrated through extensive computational experiments on benchmark datasets and a real-world application case.

How to Specify a Reference Point in Hypervolume Calculation for Fair Performance Comparison

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.

Guest Editorial Special Issue on Fuzzy Techniques in Financial Modeling and Simulation

The papers in this special section focus on the use of fuzzy techniques and logic for use in financial modeling and simulation. Computational intelligence has attracted a significant and increasing interest from the financial engineering community, and an emerging interest from analytical economics groups. The bar has been raised with the revision of regulations, and the required compliance and risk management. The new rules should be implemented through new processes and supported by developing new computational tools. Computational systems, capturing sentiments, preferences, behavior, and beliefs, are becoming indispensable in financial applications and desirable in economic analysis. They address problems in the classification of credit worthiness and fraud detection, contribute to the analysis and pricing of financial instruments, and effectively support portfolio optimization and investment analysis. They are instrumental in the design of market mechanisms and contagion mechanisms, and are contributing to the simulation of micro- and macro-economic processes. The armory of fuzzy techniques is capable of addressing challenges encountered in financial engineering and analytical economics. Fuzzy logic can effectively describe and incorporate expertsź intuition, market participantsź preferences, and economic agentsź behavior, thus reaching beyond the capabilities of probabilistic models. The objective of this special issue is to bring together the most recent advances in the design and application of fuzzy approaches to real problems in financial engineering and analytical economics.

A Decomposition-Based Evolutionary Algorithm for Multi-modal Multi-objective Optimization

This paper proposes a novel decomposition-based evolutionary algorithm for multi-modal multi-objective optimization, which is the problem of locating as many as possible (almost) equivalent Pareto optimal solutions. In the proposed method, two or more individuals can be assigned to each decomposed subproblem to maintain the diversity of the population in the solution space. More precisely, a child is assigned to a subproblem whose weight vector is closest to its objective vector, in terms of perpendicular distance. If the child is close to one of individuals that have already been assigned to the subproblem in the solution space, the replacement selection is performed based on their scalarizing function values. Otherwise, the child is newly assigned to the subproblem, regardless of its quality. The effectiveness of the proposed method is evaluated on seven problems. Results show that the proposed algorithm is capable of finding multiple equivalent Pareto optimal solutions.

Use of Two Reference Points in Hypervolume-Based Evolutionary Multiobjective Optimization Algorithms

Recently it was reported that the location of a reference point has a dominant effect on the optimal distribution of solutions for hypervolume maximization when multiobjective problems have inverted triangular Pareto fronts. This implies that the use of an appropriate reference point is indispensable when hypervolume-based EMO (evolutionary multiobjective optimization) algorithms are applied to such a problem. However, its appropriate reference point specification is difficult since it depends on various factors such as the shape of the Pareto front (e.g., triangular, inverted triangular), its curvature property (e.g., linear, convex, concave), the population size, and the number of objectives. To avoid this difficulty, we propose an idea of using two reference points: one is the nadir point, and the other is a point far away from the Pareto front. In this paper, first we demonstrate that the effect of the reference point is strongly problem-dependent. Next we propose an idea of using two reference points and its simple implementation. Then we examine the effectiveness of the proposed idea by comparing two hypervolume-based EMO algorithms: one with a single reference point and the other with two reference points.