ESLA: a new surrogate-assisted single-loop reliability-based design optimization technique

Abstract

In this paper, we address the formulation of a novel scheme for reliability-based design optimization, in which the design optimization problem is characterized by constraints that must be met with a certain probability. Assessment of the aforementioned is typically referred to as reliability analysis. Conventional methods rely on sampling approaches or by reformulating the problem as a two-level optimization that requires gradient or Hessian information of the constraints to obtain a trustworthy solution. However, the computational cost of such methods makes them often impractical. To overcome the aforementioned, a surrogate-assisted asymptotic reliability analysis (SARA) is presented that makes use of surrogate-derived gradient and Hessian information. The sub-optimization problem is reformulated as a set of constraints using the Karush-Kuhn-Tucker conditions and fitted in an efficient global optimization-like setting through the formulation of the reliability-based expected improvement (RBEI), obtaining the novel efficient single-loop approach (ESLA). The method is tested on a series of test cases which prove the effectiveness of the novel scheme.

Appendix: : Derivation of the first and second derivatives of the BLUP and MSPE

The derivation of the best linear unbiased prediction (BLUP) and the corresponding mean square predictive error (MSPE) are considered well-known and well-documented and will not be examined here.

For the derivation of the first and second derivatives of the BLUP and their respective MSPEs, we make use of a multi-variable formulation of the difference quotient (sometimes also referred to as Newton’s quotient or Fermat’s quotient) following McHutchon’s approach (McHutchon 2013). Consider the evaluation of the surrogate model at two test point locations:

$$ \begin{array}{@{}rcl@{}} \mathcal{Y}(\textbf{x}_{*})&=&\text{Y}(\textbf{x}_{*})+z_{*} \\ \mathcal{Y}(\textbf{x}_{*}+\boldsymbol{\delta})&=&\text{Y}(\textbf{x}_{*}+\boldsymbol{\delta})+z_{\delta} \end{array} $$

such that (z_∗,z_δ) takes on the form of a multi-variate Gaussian distribution P(z_∗,z_δ) of which the covariance function is given by:

$$ \left[\begin{array}{cc} \mathbb{V}[z_{*}] & \mathbb{C}[z_{\delta},z_{*}] \\ \mathbb{C}[z_{*},z_{\delta}] & \mathbb{V}[z_{\delta}] \end{array}\right] $$

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and of which the components (with $\mathbb {C}[z_{*},z_{\delta }]$ the covariance) are given by:

$$ \begin{array}{@{}rcl@{}} \mathbb{C}[z_{*},z_{\delta}]&=& \sigma^{2}\left\{1-\boldsymbol{\psi}(\boldsymbol{x}_{*})\boldsymbol{{\varPsi}}^{-1}\boldsymbol{\psi}(\textbf{x}_{*}+\boldsymbol{\delta})^{T}\right.\\ &&{}+\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}\boldsymbol{\psi}(\textbf{x}_{*})-\boldsymbol{f}(\textbf{x}_{*})\right]^{T}\cdot\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}\mathbf{F}\right]^{-1}\\ &&\left.\cdot\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}\boldsymbol{\psi}(\textbf{x}_{*}+\boldsymbol{\delta})-\boldsymbol{f}(\textbf{x}_{*}+\boldsymbol{\delta})\right]\right\} \end{array} $$

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such that

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathcal{Y}_{*}}{\partial \textbf{x}_{*}} &=&\lim\limits_{\boldsymbol{\delta}\to\boldsymbol{0}} \frac{\mathcal{Y}(\textbf{x}_{*}+\boldsymbol{\delta})-\mathcal{Y}(\textbf{x}_{*})}{\textbf{x}_{*}+\boldsymbol{\delta}-\textbf{x}_{*}} \\ &=&\lim\limits_{\boldsymbol{\delta}\to\boldsymbol{0}} \frac{\text{Y}(\textbf{x}_{*}+\boldsymbol{\delta})+z_{\delta}-\text{Y}(\textbf{x}_{*}-z_{*})}{\boldsymbol{\delta}} \\ &=&\lim\limits_{\boldsymbol{\delta}\to\boldsymbol{0}} \frac{\text{Y}(\textbf{x}_{*}+\boldsymbol{\delta})-\text{Y}(\textbf{x}_{*})}{\boldsymbol{\delta}}+\lim\limits_{\boldsymbol{\delta}\to\boldsymbol{0}} \frac{z_{\delta}-z_{*}}{\boldsymbol{\delta}} \\ &=&\frac{\partial \text{Y}_{*}}{\partial \textbf{x}_{*}}+\lim\limits_{\boldsymbol{\delta}\to\boldsymbol{0}} \frac{z_{\delta}-z_{*}}{\boldsymbol{\delta}} \end{array} $$

From these results, the derivative of the BLUP can be determined.

$$ \begin{array}{@{}rcl@{}} \frac{\partial \text{Y}(\textbf{x})}{\partial \text{x}_{i}}=& \frac{\partial\boldsymbol{f}(\textbf{x})}{\partial \text{x}_{i}}^{T}\boldsymbol{\beta}+\frac{\partial \boldsymbol{\psi}(\textbf{x})}{\partial \text{x}_{i}}^{T}\boldsymbol{\Uppsi}^{-1}(\textbf{y}-\mathbf{F}\boldsymbol{\beta}) \end{array} $$

(62)

In a similar manner, the MSPE of the derivative is derived.

$$ \begin{array}{@{}rcl@{}} \mathbb{V}\left[\underset{\boldsymbol{\delta}\to\boldsymbol{0}}{\lim}\frac{w_{\delta}-w_{*}}{\boldsymbol{\delta}}\right]{}&=&{} \lim\limits_{\boldsymbol{\delta}\to\boldsymbol{0}}\frac{1}{\boldsymbol{\delta}^{2}}{}\left\{\mathbb{V}[w_{\delta}]+{}\mathbb{V}[w_{*}]-\mathbb{C}[w_{\delta},w_{*}]\right.\\ &&{}\left.-\mathbb{C}[w_{*},w_{\delta}]\right\}= \lim\limits_{\boldsymbol{\delta}\to\boldsymbol{0}}\frac{\sigma^{2}}{\boldsymbol{\delta}^{2}}\left\{(\boldsymbol{\psi}(\textbf{x}_{*}+\boldsymbol{\delta})-\boldsymbol{\psi}(\textbf{x}_{*}))^{T}\boldsymbol{{\varPsi}}^{-1}\right. \\ &&{}+((\boldsymbol{\psi}(\textbf{x}_{*}+\boldsymbol{\delta})-\boldsymbol{\psi}(\textbf{x}_{*})))\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}(\boldsymbol{\psi}(\textbf{x}_{*}+\boldsymbol{\delta})-\boldsymbol{\psi}(\textbf{x}_{*}))\right.\\ &&{}\left.-\left( \boldsymbol{f}(\textbf{x}_{*}+\boldsymbol{\delta})-\boldsymbol{f}(\textbf{x}_{*})\right)\right]^{T}\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}\mathbf{F}\right]^{-1}\\ &&{}\cdot{}\left.\left[{}\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}{}\cdot{}(\boldsymbol{\psi}(\textbf{x}_{*}+\boldsymbol{\delta}){}-{}\boldsymbol{\psi}(\textbf{x}_{*})){}-{}(\boldsymbol{f}(\textbf{x}_{*}{}+{}\boldsymbol{\delta}){}-{}\boldsymbol{f}(\textbf{x}_{*}))\right]\right\} \\ &&{}= \sigma^{2}\left\{\frac{\partial \psi}{\partial \textbf{x}_{*}}^{T}\boldsymbol{{\varPsi}}^{-1}\frac{\partial \psi}{\partial \textbf{x}_{*}}+\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}\frac{\partial \psi}{\partial \textbf{x}_{*}}^{T}-\frac{\partial f}{\partial \textbf{x}_{*}}\right]^{T}\right. \\ &&{}\left.\cdot\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}\mathbf{F}\right]^{-1}\cdot\left[\mathbf{F}^{T}\boldsymbol{{\varPsi}}^{-1}\frac{\partial \psi}{\partial \textbf{x}_{*}}^{T}-\frac{\partial f}{\partial \textbf{x}_{*}}\right]\right\} \end{array} $$

(63)

The same approach can be repeated to obtain the second derivatives of the BLUP and their respective MSPE.