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

1.1 (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.

1.2 (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.

1.3 (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