Ryoji Tanabe

Kenji Kitamura, Ryoji Tanabe

Stopping criteria automatically determine when to stop an evolutionary algorithm, so as not to waste function evaluations on a stagnant population. Although stopping criteria play an important role in real-world applications, they have attracted little attention in the evolutionary multi-objective optimization (EMO) community. In fact, new stopping criteria for EMO have been rarely developed in recent years. One reason for the stagnation in developing stopping criteria for EMO is a lack of effective benchmarking methodologies. To address this issue, this paper proposes (i) a performance measure of stopping criteria for EMO and (ii) a file-based benchmarking approach. This paper also proposes (iii) a data representation method that effectively stores population states in text files. (i) The proposed measure represents the performance of stopping criteria as a single scalar value, making comparison easy. (ii) The proposed file-based approach not only simplifies the benchmarking process but also facilitates reproducibility. (iii) The proposed data representation method addresses the issue of file size in (ii). We demonstrate the effectiveness of our three contributions (i)--(iii) by benchmarking five representative stopping criteria for EMO.

Ryuichi Mogami, Ryoji Tanabe

Preference-based evolutionary multi-objective optimization (PBEMO) aims to approximate a region of interest (ROI) defined by the preference information from a decision maker (DM). Although objective functions in real-world applications typically have different scales, the issue of how to define the ROI in such problems has been overlooked in the literature. In fact, it has not been standardized in the EMO community whether the ROI should be defined in the unnormalized objective space or in the normalized objective space. In this context, this paper investigates the effects of objective normalization on ROIs. First, this paper shows that two ROIs defined in the unnormalized and normalized objective spaces can differ significantly for problems with differently scaled objectives. Then, we demonstrate that ROIs defined in the normalized objective space are highly difficult to approximate even on problems with equally scaled objectives because of poor approximations of the ideal and nadir points. In contrast, we show that ROIs defined in the unnormalized objective space are much easier to approximate than those defined in the normalized objective space.

Takushi Yoshikawa, Ryoji Tanabe

Feature-based offline algorithm selection has shown its effectiveness in a wide range of optimization problems, including the black-box optimization problem. An algorithm selection system selects the most promising optimizer from an algorithm portfolio, which is a set of pre-defined optimizers. Thus, algorithm selection requires a well-constructed algorithm portfolio consisting of efficient optimizers complementary to each other. Although construction methods for the fixed-target setting have been well studied, those for the fixed-budget setting have received less attention. Here, the fixed-budget setting is generally used for computationally expensive optimization, where a budget of function evaluations is small. In this context, first, this paper points out some undesirable properties of experimental setups in previous studies. Then, this paper argues the importance of considering the number of function evaluations used in the sampling phase when constructing algorithm portfolios, whereas the previous studies ignored that. The results show that algorithm portfolios constructed by our approach perform significantly better than those by the previous approach.

Ryoji Tanabe

Feature-based algorithm selection aims to automatically find the best one from a portfolio of optimization algorithms on an unseen problem based on its landscape features. Feature-based algorithm selection has recently received attention in the research field of black-box numerical optimization. However, there is still room for analysis of algorithm selection for black-box optimization. Most previous studies have focused only on whether an algorithm selection system can outperform the single-best solver in a portfolio. In addition, a benchmarking methodology for algorithm selection systems has not been well investigated in the literature. In this context, this paper analyzes algorithm selection systems on the 24 noiseless black-box optimization benchmarking functions. First, we demonstrate that the first successful performance measure is more reliable than the expected runtime measure for benchmarking algorithm selection systems. Then, we examine the influence of randomness on the performance of algorithm selection systems. We also show that the performance of algorithm selection systems can be significantly improved by using sequential least squares programming as a pre-solver. We point out that the difficulty of outperforming the single-best solver depends on algorithm portfolios, cross-validation methods, and dimensions. Finally, we demonstrate that the effectiveness of algorithm portfolios depends on various factors. These findings provide fundamental insights for algorithm selection for black-box optimization.

Ryoji Tanabe

Although exploratory landscape analysis (ELA) has shown its effectiveness in various applications, most previous studies focused only on low- and moderate-dimensional problems. Thus, little is known about the scalability of the ELA approach for large-scale optimization. In this context, first, this paper analyzes the computational cost of features in the flacco package. Our results reveal that two important feature classes (ela_level and ela_meta) cannot be applied to large-scale optimization due to their high computational cost. To improve the scalability of the ELA approach, this paper proposes a dimensionality reduction framework that computes features in a reduced lower-dimensional space than the original solution space. We demonstrate that the proposed framework can drastically reduce the computation time of ela_level and ela_meta for large dimensions. In addition, the proposed framework can make the cell-mapping feature classes scalable for large-scale optimization. Our results also show that features computed by the proposed framework are beneficial for predicting the high-level properties of the 24 large-scale BBOB functions.

Ryoji Tanabe, Alex Fukunaga

Many Differential Evolution (DE) algorithms with various parameter control methods (PCMs) have been proposed. However, previous studies usually considered PCMs to be an integral component of a complex DE algorithm. Thus the characteristics and performance of each method are poorly understood. We present an in-depth review of 24 PCMs for the scale factor and crossover rate in DE and a large scale benchmarking study. We carefully extract the 24 PCMs from their original, complex algorithms and describe them according to a systematic manner. Our review facilitates the understanding of similarities and differences between existing, representative PCMs. The performance of DEs with the 24 PCMs and 16 variation operators is investigated on 24 black-box benchmark functions. Our benchmarking results reveal which methods exhibit high performance when embedded in a standardized framework under 16 different conditions, independent from their original, complex algorithms. We also investigate how much room there is for further improvement of PCMs by comparing the 24 methods with an oracle-based model, which can be considered to be a conservative lower bound on the performance of an optimal method.

Ryoji Tanabe, Alex Fukunaga

We consider how an (almost) optimal parameter adaptation process for an adaptive DE might behave, and compare the behavior and performance of this approximately optimal process to that of existing, adaptive mechanisms for DE. An optimal parameter adaptation process is an useful notion for analyzing the parameter adaptation methods in adaptive DE as well as other adaptive evolutionary algorithms, but it cannot be known generally. Thus, we propose a Greedy Approximate Oracle method (GAO) which approximates an optimal parameter adaptation process. We compare the behavior of GAODE, a DE algorithm with GAO, to typical adaptive DEs on six benchmark functions and the BBOB benchmarks, and show that GAO can be used to (1) explore how much room for improvement there is in the performance of the adaptive DEs, and (2) obtain hints for developing future, effective parameter adaptation methods for adaptive DEs.

Ryoji Tanabe, Hisao Ishibuchi

An unbounded external archive (UEA), which stores all nondominated solutions found during the search process, is frequently used to evaluate the performance of multi-objective evolutionary algorithms (MOEAs) in recent studies. A recent benchmarking study also shows that decomposition-based MOEA (MOEA/D) is competitive with state-of-the-art MOEAs when the UEA is incorporated into MOEA/D. However, a parameter study of MOEA/D using the UEA has not yet been performed. Thus, it is unclear how control parameter settings influence the performance of MOEA/D with the UEA. In this paper, we present an analysis of control parameters of MOEA/D under two performance evaluation scenarios. One is a final population scenario where the performance assessment of MOEAs is performed based on all nondominated solutions in the final population, and the other is a reduced UEA scenario where it is based on a pre-specified number of selected nondominated solutions from the UEA. Control parameters of MOEA/D investigated in this paper include the population size, scalarizing functions, and the penalty parameter of the penalty-based boundary intersection (PBI) function. Experimental results indicate that suitable settings of the three control parameters significantly depend on the choice of an optimization scenario. We also analyze the reason why the best parameter setting is totally different for each scenario.

Ryoji Tanabe, Hisao Ishibuchi

Multi-modal multi-objective optimization is to locate (almost) equivalent Pareto optimal solutions as many as possible. Some evolutionary algorithms for multi-modal multi-objective optimization have been proposed in the literature. However, there is no efficient method for multi-modal many-objective optimization, where the number of objectives is more than three. To address this issue, this paper proposes a niching indicator-based multi-modal multi- and many-objective optimization algorithm. In the proposed method, the fitness calculation is performed among a child and its closest individuals in the solution space to maintain the diversity. The performance of the proposed method is evaluated on multi-modal multi-objective test problems with up to 15 objectives. Results show that the proposed method can handle a large number of objectives and find a good approximation of multiple equivalent Pareto optimal solutions. The results also show that the proposed method performs significantly better than eight multi-objective evolutionary algorithms.

Ryoji Tanabe, Hisao Ishibuchi

Although synthetic test problems are widely used for the performance assessment of evolutionary multi-objective optimization algorithms, they are likely to include unrealistic properties which may lead to overestimation/underestimation. To address this issue, we present a multi-objective optimization problem suite consisting of 16 bound-constrained real-world problems. The problem suite includes various problems in terms of the number of objectives, the shape of the Pareto front, and the type of design variables. 4 out of the 16 problems are multi-objective mixed-integer optimization problems. We provide Java, C, and Matlab source codes of the 16 problems so that they are available in an off-the-shelf manner. We examine an approximated Pareto front of each test problem. We also analyze the performance of six representative evolutionary multi-objective optimization algorithms on the 16 problems. In addition to the 16 problems, we present 8 constrained multi-objective real-world problems.

Ryoji Tanabe

Since the scale factor and the crossover rate significantly influence the performance of differential evolution (DE), parameter adaptation methods (PAMs) for the two parameters have been well studied in the DE community. Although PAMs can sufficiently improve the effectiveness of DE, PAMs are poorly understood (e.g., the working principle of PAMs). One of the difficulties in understanding PAMs comes from the unclarity of the parameter space that consists of the scale factor and the crossover rate. This paper addresses this issue by analyzing adaptive parameter landscapes in PAMs for DE. First, we propose a concept of an adaptive parameter landscape, which captures a moment in a parameter adaptation process. For each iteration, each individual in the population has its adaptive parameter landscape. Second, we propose a method of analyzing adaptive parameter landscapes using a 1-step-lookahead greedy improvement metric. Third, we examine adaptive parameter landscapes in PAMs by using the proposed method. Results provide insightful information about PAMs in DE.