Non-parametric stochastic subset optimization for design problems with reliability constraints

Abstract

The Non-Parametric Stochastic Subset Optimization (NP-SSO) is a recently developed algorithm appropriate for optimization problems that use reliability criteria as objective function and involve computationally expensive numerical models for the engineering system under consideration. This paper discusses its extension to reliability-based design optimization (RBDO) applications involving reliability criteria as a design constraint. The foundation of NP-SSO is the formulation of an augmented problem where the design variables are artificially considered as uncertain. In this context, the reliability of the engineering system is proportional to an auxiliary probability density function related to the design variables. NP-SSO is based on simulation of samples from this density and approximation of this reliability through kernel density estimation (KDE) using these samples. The RBDO problem is then solved using this approximation for evaluating the reliability constraints over the entire design domain and identifying the feasible region satisfying them. To improve computational efficiency, an iterative approach is proposed; at the end of each iteration, a new reduced search space is identified with reliability satisfying relaxed constraints, until the algorithm converges to the feasible design domain satisfying the desired constraints. A second refinement stage after initial convergence is also proposed to further improve the accuracy of the identified feasible region. A non-parametric characterization of the search space using a framework based on multivariate boundary KDE and support vector machine is established. To further improve the efficiency of the stochastic sampling stage, an adaptive selection of the number of samples required for the KDE approximation is proposed.

Appendix A: Details on multivariate boundary kernels

If n _s samples {x _F } are available for x from p(x| F _I ) with x ^j _F,i denoting the j ^th sample of the i ^th design variable, the KDE-MBK estimator is given by utilizing a corrected kernel ${\underline{K}}_m$

$$ \begin{array}{l}\tilde{p}\left(\mathbf{x}\Big|{F}_I\right)=\frac{1}{n_s}{\displaystyle \sum_{j=1}^{n_s}}{\underline{K}}_m\left(\mathbf{x}-{\mathbf{x}}_F^j\right)\\ {}{\underline{K}}_m\left(\mathbf{x}-{\mathbf{x}}_F^j\right)=\frac{K_m\left(\mathbf{x}-{\mathbf{x}}_F^j\right)-{K}_m\left(\mathbf{x}-{\mathbf{x}}_F^j\right){\left[\mathbf{x}-{\mathbf{x}}_F^j\right]}^T{D}^{-1}\left(\mathbf{x}\right){c}_1\left(\mathbf{x}\right)}{c_0\left(\mathbf{x}\right)-{c}_1^T\left(\mathbf{x}\right){D}^{-1}\left(\mathbf{x}\right){c}_1\left(\mathbf{x}\right)},\end{array} $$

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where the uncorrected multivariate kernel K _m is the product kernel

$$ {K}_m\left(\mathbf{x}-{\mathbf{x}}_F^j\right)={\displaystyle \prod_{i=1}^{n_x}}\frac{1}{w_i}K\left(\frac{x_i-{x}_{F,i}^j}{w_i}\right), $$

(22)

with K(.) the chosen univariate kernel, w _i the bandwidth parameter for the i ^th design variable defining the spread of the kernel, and constant c ₀(x)_, vector ${c}_1\left(\mathbf{x}\right)={\left[\begin{array}{ccc}\hfill {c}_{11}\left(\mathbf{x}\right)\hfill & \hfill \cdots \hfill & \hfill {c}_{1{n}_x}\left(\mathbf{x}\right)\hfill \end{array}\right]}^T$, and matrix D(x) with elements d _kl (x), given, respectively, by:

$$ \begin{array}{l}{c}_0\left(\mathbf{x}\right)={\displaystyle {\int}_{\varDelta_I}{K}_m\left(\mathbf{x}-\mathbf{y}\right)}d\mathbf{y},\ {c}_{1i}\left(\mathbf{x}\right)={\displaystyle {\int}_{\varDelta_I}{K}_m\left(\mathbf{x}-\mathbf{y}\right)}\left({x}_i-{y}_i\right)d\mathbf{y}\\ {}{d}_{kl}\left(\mathbf{x}\right)={\displaystyle {\int}_{\varDelta_I}{K}_m\left(\mathbf{x}-\mathbf{y}\right)}\left({x}_k-{y}_k\right)\left({x}_l-{y}_l\right)d\mathbf{y}.\end{array} $$

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Here Δ _Ι is the part of the support region S(x) for the kernel K _m (x) that also belongs to I. If the whole part of Δ _Ι is inside I then x is an interior point and no correction is necessary ${\underline{K}}_m={K}_m$; otherwise, it belongs to the boundary region and estimation of ${\underline{K}}_m$ requires the variables (correction coefficients) in (23), which for complex boundaries can be performed using Monte Carlo (MC) integration (discussed next). For the univariate kernel, the Epanechnikov Kernel is chosen in this study

$$ \begin{array}{l}K(t)=\frac{3}{4}\left(1-{t}^2\right)\kern1em \mathrm{if}\kern0.75em -1\le t\le 1\\ {}=0\kern0.75em \mathrm{else}\end{array} $$

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with bandwidth characteristics the ones derived in (Jia and Taflanidis 2014)

$$ {w}_i={\left[25{\left(\frac{3}{5}\right)}^{n_x}{\left(2\sqrt{\uppi}\right)}^{n_x}\right]}^{1/\left({n}_x+4\right)}{\left[\frac{4}{\left({n}_x+2\right){n}_s}\right]}^{1/\left({n}_x+4\right)}{\sigma}_i $$

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where σ _i corresponds to the standard deviation for the samples {x _F,i }.

The correction coefficients in (23) can be estimated in the following way using MC integration with a uniform proposal density in Δ _Ι : first, generate n ^coeff _u,x uniform samples {x ^j _u ; j = 1, …, n ^coeff _u,x } in the support region S(x) for x, which for the Epanechnikov Kernel is a hyper-rectangle centred at x and with width 2w _i in the i ^th dimension and a volume of ${V}_c={2}^{n_x}{\displaystyle {\prod}_{i=1}^{n_x}{w}_i}$. If the domain/boundary is characterized by a SVM, then use this SVM to classify the n ^coeff _u,x samples and identify the n ^coeff _in,x uniform samples {x ^j _o ; j = 1, …, n ^coeff _in,x } that are inside Δ _Ι and estimate its volume, i.e., V _in  = n ^coeff _in,x /n ^coeff _u,x  ⋅ V _c . Based on this information (i.e., uniform samples within Δ _I ), calculate the required coefficients as:

$$ \begin{array}{l}{c}_0\left(\mathbf{x}\right)\approx {V}_{in}/{n}_{in,\mathbf{x}}^{coeff}{\displaystyle {\sum}_{j=1}^{n_{in,\mathbf{x}}^{coeff}}{K}_m\left(\mathbf{x}-{\mathbf{x}}_o^j\right)}\\ {}{c}_{1i}\left(\mathbf{x}\right)\approx {V}_{in}/{n}_{in,\mathbf{x}}^{coeff}{\displaystyle {\sum}_{j=1}^{n_{in,\mathbf{x}}^{coeff}}{K}_m\left(\mathbf{x}-{\mathbf{x}}_o^j\right)}\left({x}_i-{x}_{o,i}^j\right)\\ {}{d}_{kl}\left(\mathbf{x}\right)\approx {V}_{in}/{n}_{in,\mathbf{x}}^{coeff}{\displaystyle {\sum}_{j=1}^{n_{in,\mathbf{x}}^{coeff}}{K}_m\left(\mathbf{x}-{\mathbf{x}}_o^j\right)}\left({x}_k-{x}_{o,k}^j\right)\left({x}_l-{x}_{o,l}^j\right)\end{array} $$

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