Surrogate Assisted High-dimensional Multi-objective Evolutionary Optimization Methods

Some mechanical design problems,such as vehicle crashworthiness and the design of crude oil distillation units,need to optimize multiple objective functions simultaneously,in which each objective is expensive to evaluate.Evolutionary optimization algorithms have achieved more and more applications in mechanical design optimizations due to their good capability of global convergence and easily implementation.However,many expensive objective evaluations are normally required by the evolutionary optimization algorithms to obtain the global optimum,preventing their applications in those mechanical optimization problems with expensive objective functions.In recent years,surrogate-assisted evolutionary optimization algorithms are common methods for solving expensive optimization problems.However,the computational complexity will increase when the complexity of optimization problems increases.For example,the difficulty of training a model will be increased with the increase of the decision variables,and the selection pressure will be reduced,resulting in the difficulty to distinguish solutions when the dimension of the objective space increases.Based on the existing surrogate-assisted multi-objective evolutionary algorithms,we go into the model management strategy and propose new surrogate-assisted multi-objective algorithms.A number of benchmark problems and some real-world multi-objective mechanical optimization problems will be used to evaluate the effectiveness and feasibility of the proposed method.The main contents of this article are given as follows:1.The radial basis function model is good for fit nonlinear and higher-order problems.Furthermore,the complexity to train the RBF surrogate model is low.Thus,we propose to train an RBF surrogate model for each objective function of expensive many-objective problems.A model-based optimization is conducted for finding a good population.The infill criterion is determined adaptively by the convergence performance of the current population to choose solutions for expensive objective evaluation either with the best approximated objective values or with the maximum approximation uncertainty.In the proposed method,the non-dominant sorting method is used to measure the degree of convergence,and the Euclidean distances of an individual to its nearest sample and its nearest samples are used to measure the approximation uncertainty of this solution.In addition,the initial population of optimization is composed of solutions with the best convergence and diversity performance for each optimization search.The effectiveness of the method is verified by conducting experiments on some benchmark problems.2.Different surrogate models often have their unique advantages and disadvantages.Thus,it is difficult to select appropriate surrogate models for assisting evolutionary optimization algorithms where the problem characteristics are unknown in advance.In this paper,we train multiple surrogate models for each objective function of an expensive multi-objective problem and model-based optimization is conducted.Both the variance of the approximation values from different models and Euclidean distance in the decision space between the solution and the samples are used to calculate the approximation uncertainty of the solution.Then an infill criterion is determined according to the degree of change in the distance from the population to the non-dominated front,and some solutions will be adaptively selected for expensive evaluation.Furthermore,some non-dominated solutions will be added to the population of the next optimization to ensure the convergence of the population.The effectiveness of the algorithm will be verified by comparing the experimental results on benchmark problems with some state-of-the-art algorithms.3.For different types of optimization problems,the ensemble of multiple surrogate models can be a strong learner and has better robustness than one surrogate model.Therefore,two RBF models with different kernel functions are used as an ensemble of surrogate models for solving expensive multi-objective problems.A two-stage sampling strategy is proposed for model management.In the first stage,a solution for expensive evaluation is selected according to the approximated value and the approximation variance of the ensemble.In the second stage,the solution with the maximum uncertainty based on the variance of the approximation resulted from model updating is selected.In addition,the approximate errors of the surrogate models gradually accumulate with the increase of objectives.Thus,a population correction strategy is proposed to prevent the population from deviating from the correct search direction.In the population correction strategy,the objective values of solutions in the population before and after optimization,respectively,are re-approximated using the updated surrogate model,and solutions with better approximated values will be kept to be the initial population of the next iteration.The effectiveness of the algorithm is verified by the experimental results of the benchmark problems.4.The proposed methods are applied to three mechanical expensive optimization design problems to evaluate their effectiveness further.