Gaussian mixture model for robust design optimization of planar steel frames

Abstract

A new method is presented for an application of the Gaussian mixture model (GMM) to a multi-objective robust design optimization (RDO) of planar steel frame structures under aleatory (stochastic) uncertainty in material properties, external loads, and discrete design variables. Uncertainty in the discrete design variables is modeled in the wide range between the smallest and largest values in the catalog of the cross-sectional areas. A weighted sum of Gaussians is statistically trained based on the sampled training data to capture an underlying joint probability distribution function (PDF) of random input variables and the corresponding structural response. A simple regression function for predicting the structural response can be found by extracting the information from a conditional PDF, which is directly derived from the captured joint PDF. A multi-objective RDO problem is formulated with three objective functions, namely, the total mass of the structure, and the mean and variance values of the maximum inter-story drift under some constraints on design strength and serviceability requirements. The optimization problem is solved using a multi-objective genetic algorithm utilizing the trained GMM for calculating the statistical values of objective and constraint functions to obtain Pareto-optimal solutions. Since the three objective functions are highly conflicting, the best trade-off solution is desired and found from the obtained Pareto-optimal solutions by performing fuzzy-based compromise programming. The robustness and feasibility of the proposed method for finding the RDO of planar steel frame structures with discrete variables are demonstrated through two design examples.

Appendix. Gradient and Hessian of GMMs

Let x = [x₁, x₂, …, x_D]^T ∈ ℝ^D be D-dimensional vectors generated from a D-variate Gaussian ${p}_{\mathbf{X}}\left(\mathbf{x}\right)\sim {\mathcal{N}}_D\left(\boldsymbol{\upmu}, \boldsymbol{\Sigma} \right)$, such that

$$ {p}_{\mathbf{X}}\left(\mathbf{x}\right)=\frac{1}{{\left(2\pi \right)}^{D/2}{\left|\boldsymbol{\Sigma} \right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(\mathbf{x}-\boldsymbol{\upmu} \right)}^T{\boldsymbol{\Sigma}}^{-1}\left(\mathbf{x}-\boldsymbol{\upmu} \right)\right] $$

(45)

Let Σ = UΛU^T be the singular value decomposition of the covariance matrix Σ, where U is an orthogonal matrix and Λ is a non-singular diagonal matrix with λ_d at the dth diagonal component. Let y = U^T(x − μ) ∈ ℝ^D so that ∂y_e/∂x_d = u_de, where e ∈ {1, 2, …, D}, and u_de is the (d, e)th element of U. Taking the first derivatives of p_X(x), we have

$$ {\displaystyle \begin{array}{c}\frac{\partial p}{\partial {x}_d}={p}_{\mathbf{x}}\left(\mathbf{x}\right)\frac{\partial }{\partial {x}_d}\left[-\frac{1}{2}{\left(\mathbf{x}-\boldsymbol{\upmu} \right)}^T{\boldsymbol{\Sigma}}^{-1}\left(\mathbf{x}-\boldsymbol{\upmu} \right)\right]={p}_{\mathbf{x}}\left(\mathbf{x}\right)\frac{\partial }{\partial {x}_d}\left(-\frac{1}{2}{\mathbf{y}}^T{\boldsymbol{\Lambda}}^{-1}\mathbf{y}\right)\\ {}\kern1.2em ={p}_{\mathbf{x}}\left(\mathbf{x}\right)\sum \limits_{e=1}^D\frac{\partial }{\partial {y}_e}\left(-\frac{1}{2}\sum \limits_{d=1}^D\frac{y_d^2}{\lambda_d}\right)\frac{\partial {y}_e}{\partial {x}_d}=-{p}_{\mathbf{x}}\left(\mathbf{x}\right)\sum \limits_{e=1}^D\frac{y_e}{\lambda_e}{u}_{de}\end{array}} $$

(46)

which is the dth element of vector −p_X(x)UΛ⁻¹y, which yields the gradient of p_X(x) as

$$ \mathbf{g}=\nabla {p}_{\mathbf{X}}\left(\mathbf{x}\right)={p}_{\mathbf{X}}\left(\mathbf{x}\right){\boldsymbol{\Sigma}}^{-1}\left(\boldsymbol{\upmu} -\mathbf{x}\right) $$

(47)

Let c ∈ {1, 2, …, D}, and taking the second derivatives of p_X(x), we obtain

$$ {\displaystyle \begin{array}{c}\frac{\partial }{\partial {x}_c}\left(\frac{\partial p}{\partial {x}_d}\right)=\frac{\partial }{\partial {x}_c}\left(-{p}_{\mathbf{x}}\left(\mathbf{x}\right)\sum \limits_{e=1}^D\frac{y_e}{\lambda_e}{u}_{de}\right)=-{p}_{\mathbf{x}}\left(\mathbf{x}\right)\frac{\partial }{\partial {x}_c}\left(\sum \limits_{e=1}^D\frac{y_e}{\lambda_e}{u}_{de}\right)-\frac{\partial {p}_{\mathbf{X}}\left(\mathbf{x}\right)}{\partial {x}_c}\sum \limits_{e=1}^D\frac{y_e}{\lambda_e}{u}_{de}\\ {}=-{p}_{\mathbf{x}}\left(\mathbf{x}\right)\left(\sum \limits_{e=1}^D\frac{u_{de}}{\lambda_d}\frac{\partial {y}_e}{\partial {x}_c}\right)+{p}_{\mathbf{x}}\left(\mathbf{x}\right)\left(\sum \limits_{e=1}^D\frac{y_e}{\lambda_e}{u}_{ce}\right)\left(\sum \limits_{e=1}^D\frac{y_e}{\lambda_e}{u}_{de}\right)\\ {}=-{p}_{\mathbf{x}}\left(\mathbf{x}\right)\left(\sum \limits_{e=1}^D\frac{u_{de}}{\lambda_d}{u}_{ce}\right)+{p}_{\mathbf{x}}\left(\mathbf{x}\right)\left(\sum \limits_{e=1}^D\frac{y_d}{\lambda_e}{u}_{ce}\right)\left(\sum \limits_{e=1}^D\frac{y_d}{\lambda_e}{u}_{de}\right)\end{array}} $$

(48)

which is the (c, d)th element of the matrix −p_X(x)UΛ⁻¹U^T + [p_X(x)]⁻¹gg^T (Carreira-Perpinan 2000). Hence, the Hessian of p_X(x) is

$$ \mathbf{H}=\left(\nabla {\nabla}^T\right){p}_{\mathbf{X}}\left(\mathbf{x}\right)=-{p}_{\mathbf{X}}\left(\mathbf{x}\right){\boldsymbol{\Sigma}}^{-1}+{\left[{p}_{\mathbf{X}}\left(\mathbf{x}\right)\right]}^{-1}{\mathbf{gg}}^T={p}_{\mathbf{X}}\left(\mathbf{x}\right){\boldsymbol{\Sigma}}^{-1}\left[-\boldsymbol{\Sigma} +\left(\mathbf{x}-\boldsymbol{\upmu} \right){\left(\mathbf{x}-\boldsymbol{\upmu} \right)}^T\right]{\boldsymbol{\Sigma}}^{-1} $$

(49)

Results in (47) and (49) lead to the gradient and Hessian of the GMM $\sum \limits_{k=1}^K{\pi}_k\phi \left(\mathbf{x};{\boldsymbol{\upmu}}_{\mathbf{X},k},{\boldsymbol{\Sigma}}_{\mathbf{X},k}\right)$ as

$$ {\mathbf{g}}_m=\nabla \left[\sum \limits_{k=1}^K{\pi}_k\phi \left(\mathbf{x};{\boldsymbol{\upmu}}_{\mathbf{X},k},{\boldsymbol{\Sigma}}_{\mathbf{X},k}\right)\right]=\sum \limits_{k=1}^K{\pi}_k\phi \left(\mathbf{x};{\boldsymbol{\upmu}}_{\mathbf{X},k},{\boldsymbol{\Sigma}}_{\mathbf{X},k}\right){\boldsymbol{\Sigma}}_{\mathbf{X},k}^{-1}\left({\boldsymbol{\upmu}}_{\mathbf{X},k}-\mathbf{x}\right) $$

(50)

$$ {\mathbf{H}}_m\kern0.5em {\displaystyle \begin{array}{c}=\left(\nabla {\nabla}^T\right)\left[\sum \limits_{k=1}^K{\pi}_k\phi \left(\mathbf{x};{\boldsymbol{\upmu}}_{\mathbf{X},k},{\boldsymbol{\Sigma}}_{\mathbf{X},k}\right)\right]\\ {}=\sum \limits_{k=1}^K{\pi}_k\phi \left(\mathbf{x};{\boldsymbol{\upmu}}_{\mathbf{X},k},{\boldsymbol{\Sigma}}_{\mathbf{X},k}\right){\boldsymbol{\Sigma}}_{\mathbf{X},k}^{-1}\left[-{\boldsymbol{\Sigma}}_{\mathbf{X},k}+\left(\mathbf{x}-{\boldsymbol{\upmu}}_{\mathbf{X},k}\right){\left(\mathbf{x}-{\boldsymbol{\upmu}}_{\mathbf{X},k}\right)}^T\right]{\boldsymbol{\Sigma}}_{\mathbf{X},k}^{-1}\end{array}} $$

(51)

Since the mixing weight w_k(x) of the regression function in (13) is a fraction of two GMMs, the following quotient rules, which is applied for calculating the derivatives of f(x) = g(x)/h(x), can be utilized to obtain the gradient and Hessian of the regression function in (15) and (16):

$$ \frac{\partial f}{\partial x}=\frac{h\partial g/\partial x-g\partial h/\partial x}{h^2} $$

(52)

$$ \frac{\partial^2f}{\partial {x}^2}=\frac{\partial^2}{\partial {x}^2}\left(\frac{g}{h}\right)=\frac{\partial^2g/\partial {x}^2-2\left[\partial f/\partial x\right]{\left[\partial f/\partial h\right]}^T-f\left[{\partial}^2\mathrm{h}/\partial {x}^2\right]}{h} $$

(53)