Expensive multiobjective optimization problems pose great challenges to evolutionary algorithms due to their costly evaluation. Building cheap surrogate models to replace the expensive real models has been proved to be a practical way to reduce the number of costly evaluations. Supervised learning techniques from the community of machine learning have been widely applied to build either regressors, which approximate the fitness values of candidate solutions, or classifiers, which estimate the categories of candidate solutions. Considering the characteristics of the data produced in optimization, this article proposes a new surrogate model, called a relation model, for evolutionary multiobjective optimization. Instead of estimating the qualities of candidate solutions directly, the relation model tries to estimate the relationship between a pair of solutions $\langle \mathbf {x}, \mathbf {y}\rangle $ , i.e., $\mathbf {x}$ dominates $\mathbf {y}$ , $\mathbf {x}$ is dominated by $\mathbf {y}$ , or $\mathbf {x}$ is nondominated with $\mathbf {y}$ in the case of multiobjective optimization. To implement this idea, first a balanced training set is prepared, then a classifier is built based on the training data set to learn the relationship, and finally, the classifier with a voting-scoring strategy is applied to estimate the relationship between the candidate solutions and parent solutions. By this way, the promising candidate solutions are recognized and evaluated by the real models. The new approach is applied to three well-known benchmark suites and two real-world applications, and the experimental results suggest that the proposed method outperforms some state-of-the-art methods based on regression and classification models on the given instances.

Under mild conditions, the Pareto front (Pareto set) of a continuous m-objective optimization problem forms an (m - 1)-dimensional piecewise continuous manifold. Based on this property, this paper proposes a self-organizing multiobjective evolutionary algorithm. At each generation, a self-organizing mapping method with (m - 1) latent variables is applied to establish the neighborhood relationship among current solutions. A solution is only allowed to mate with its neighboring solutions to generate a new solution. To reduce the computational overhead, the self-organizing training step and the evolution step are conducted in an alternative manner. In other words, the self-organizing training is performed only one single step at each generation. The proposed algorithm has been applied to a number of test instances and compared with some state-of-the-art multiobjective evolutionary methods. The results have demonstrated its advantages over other approaches.

Cooperative co-evolution (CC) is an efficient and practical evolutionary framework for solving large-scale optimization problems. The performance of CC is affected by the variable decomposition. An accurate variable decomposition can help to improve the performance of CC on solving an optimization problem. The variable grouping methods usually spend many computational resources obtaining an accurate variable decomposition. To reduce the computational cost on the decomposition, we propose an efficient recursive differential grouping (ERDG) method in this paper. By exploiting the historical information on examining the interrelationship between the variables of an optimization problem, ERDG is able to avoid examining some interrelationship and spend much less computation than other recursive differential grouping methods. Our experimental results and analysis suggest that ERDG is a competitive method for decomposing large-scale continuous problems and improves the performance of CC for solving the large-scale optimization problems.

Random forest (RF) is a type of ensemble-based machine learning method that has been applied to a variety of machine learning tasks in recent years. This article proposes an evolutionary approach to generate an oblique RF for regression problems. More specifically, our method induces an oblique RF by transforming the original feature space to a new feature space through the evolutionary feature construction method. To speed up the searching process, the proposed method evaluates each set of features based on a decision tree (DT) rather than an RF. In order to obtain an RF, we archive top-performing features and corresponding trees during the search. In this way, both the features and the forest can be constructed simultaneously in a single run. The proposed evolutionary forest is applied to 117 benchmark problems with different characteristics and compared with some state-of-the-art regression methods, including several variants of the RF and gradient boosted DTs (GBDTs). The experimental results suggest that the proposed method outperforms the existing RF and GBDT methods.