A dynamic surrogate-assisted evolutionary algorithm framework for expensive structural optimization

Abstract

In the expensive structural optimization, the data-driven surrogate model has been proven to be an effective alternative to physical simulation (or experiment). However, the static surrogate-assisted evolutionary algorithm (SAEA) often becomes powerless and inefficient when dealing with different types of expensive optimization problems. Therefore, how to select high-reliability surrogates to assist an evolutionary algorithm (EA) has always been a challenging task. This study aimed to dynamically provide an optimal surrogate for EA by developing a brand-new SAEA framework. Firstly, an adaptive surrogate model (ASM) selection technology was proposed. In ASM, according to different integration criteria from the strategy pool, elite meta-models were recombined into multiple ensemble surrogates in each iteration. Afterward, a promising model was adaptively picked out from the model pool based on the minimum root of mean square error (RMSE). Secondly, we investigated a novel ASM-based EA framework, namely ASMEA, where the reliability of all models was updated in real-time by generating new samples online. Thirdly, to verify the performance of the ASMEA framework, two instantiation algorithms are widely compared with several state-of-the-art algorithms on a commonly used benchmark test set. Finally, a real-world antenna structural optimization problem was solved by the proposed algorithms. The results demonstrate that the proposed framework is able to provide a high-reliability surrogate to assist EA in solving expensive optimization problems.

Appendices

Appendix1

(1) Kriging (KRG)

A general form of the Kriging model is as follows:

$$ f(x)=g(x)+z(x) $$

(1)

where is an unknown prediction function, is a known approximate function, and is the realization of a stochastic process with a zero mean and a nonzero covariance. The (i,j)th element of the covariance matrix z(x) is given as

$$ \mathrm{Cov}\left[z\left({x}^i\right),z\left({x}^j\right)\right]=\sigma \frac{2}{s}{R}_{ij} $$

(2)

where $\sigma \frac{2}{z}$ is the process variance, and Rij is the correlation function between the ith and the jth data points. A Gaussian function is used as the correlation function in this paper, defined by

$$ {R}_{ij}=R\left({X}^i,{X}_j\right)=\exp \left\{-\sum \limits_{k-1}^n{\theta}_k{\left({x}_k^i-{x}_k^i\right)}^2\right\} $$

(3)

where θ_k is distinct for each dimension, and these unknown parameters are generally obtained by solving a nonlinear optimization problem. In this paper, the order of the global polynomial trend function is specified to be zero.

(2) Polynomial response surface (PRS)

It is relatively easy to establish a polynomial regression model using least squares. In general, the second-order polynomial regression model can solve most problems well, which can be expressed as

$$ {y}_i\left({x}_i\right)={\beta}_o+\sum \limits_{j=1}^k{\beta}_j{x}_{ij}+\sum \limits_{j=1}^k{\beta}_j{x}_{ij}^2+\sum \limits_{j_1=1}^k\sum \limits_{\underset{J_2\ne {j}_1}{j_2=1}}^k{\beta}_{j_1{j}_2}{x}_{i,{j}_1}{x}_{i,{j}_2}+{\in}_i $$

(4)

where x_i(i = 1, 2, …, n) is design variable. yi is the corresponding predictive response. k is the number of variables, and is the fitting error. β₁ is the parameter to be determined, which can be calculated by the least square method

$$ \beta ={\left({\mathbf{F}}^{\mathrm{T}}\mathbf{F}\right)}^{-1}{\mathbf{F}}^{\mathrm{T}}y $$

(5)

where F is a Gram-matrix based on sample x.

(3) Radial basis function (RBF)

RBF has been developed for the interpolation of scattered multivariate data. The model is defined by

$$ \overset{\sim }{f}(x)=\sum \limits_{i=1}^N{\sigma}_i\psi \left(\left\Vert x-{x}_i\right\Vert \right) $$

(6)

where N denotes the number of training points, is the unknown coefficients, and is the multi-quadric function. The multi-quadric function is defined as

$$ \psi (r)=\sqrt{r^2+{c}^2} $$

(7)

where r = ‖x − x_i‖ is the Euclidean distance of a point x from a given data point xi, and the parameter c > 0 is a constant in the multi-quadric.

Appendix 2

Appendix 3