Coupling the cross-entropy with the line sampling method for risk-based design optimization

Abstract

An algorithm for risk-based optimization (RO) of engineering systems is proposed, which couples the Cross-entropy (CE) optimization method with the Line Sampling (LS) reliability method. The CE-LS algorithm relies on the CE method to optimize the total cost of a system that is composed of the design and operation cost (e.g., production cost) and the expected failure cost (i.e., failure risk). Guided by the random search of the CE method, the algorithm proceeds iteratively to update a set of random search distributions such that the optimal or near-optimal solution is likely to occur. The LS-based failure probability estimates are required to evaluate the failure risk. Throughout the optimization process, the coupling relies on a local weighted average approximation of the probability of failure to reduce the computational demands associated with RO. As the CE-LS algorithm proceeds to locate a region of design parameters with near-optimal solutions, the local weighted average approximation of the probability of failure is refined. The adaptive refinement procedure is repeatedly applied until convergence criteria with respect to both the optimization and the approximation of the failure probability are satisfied. The performance of the proposed optimization heuristic is examined empirically on several RO problems, including the design of a monopile foundation for offshore wind turbines.

Appendices

Appendix: A: Nadaraya-Watson kernel estimator

The Nadaraya-Watson kernel estimator (Nadaraya 1964; Watson 1964; Wasserman 2006) is constructed on N pairs of observations (t ₁,Y ₁),...,(t _N ,Y _N ), where a response variable Y is related to the covariate t=[t ₁,...,t _n ]^T with the following model:

$$ Y_{i}=r(\textbf{t}_{i})+\sigma(\textbf{t}_{i})\kappa_{i}; \quad i=1,...,N $$

(33)

such that r is the regression function, while σ(t _i )κ _i is the residual with location dependent variance, σ ²(t _i ).

The Nadaraya-Watson kernel estimator of r(t) is defined by:

$$ \hat{r}_{N}(\textbf{t})=\sum\limits_{i=1}^{N}w_{i}(\textbf{t})Y_{i} $$

(34)

where w _i (t) is a weight:

$$ w_{i}(\textbf{t})=\frac{K_{H}(\textbf{t}-\textbf{t}_{i})}{\sum\limits_{j=1}^{N}K_{H}(\textbf{t}-\textbf{t}_{j})} $$

(35)

with kernel function:

$$ K_{H}(\textbf{v})=\frac{1}{|H|^{1/2}}K(H^{-1/2}\textbf{v}) $$

(36)

K is a function defined to provide higher weights to observations closer to v=0, while H is a nonsingular positive definite bandwith matrix. Often the covariates are scaled so that a one-dimensional kernel with bandwidth, h>0, can be employed:

$$ K_{h}(\textbf{v})=h^{-k}K(||\textbf{v}||/h) $$

(37)

The variance of the estimator in (34) is (Wasserman 2006):

$$ \text{Var} \left\lbrack \hat{r}_{N}(\textbf{t}) \right\rbrack=\sigma^{2}(\textbf{t}) {\sum}_{i=1}^{N} {w_{i}^{2}}(\textbf{t}) $$

(38)

An estimate of σ ²(t) is evaluated based on a vector of fitted values, $\hat {\textbf {r}}_{N}=\left \lbrack \hat {r}_{N}(\textbf {t}_{1}),...,\hat {r}_{N}(\textbf {t}_{N}) \right \rbrack ^{T}$, which is calculated as:

$$ \hat{\textbf{r}}_{N}=\textbf{W}\textbf{Y} $$

(39)

where Y=[Y ₁,...,Y _N ]^T is the vector of observed response variables, while W is a N×N ’hat’ or ’smoothing’ matrix with entries W _{i j} = w _j (t _i ). Starting from the expression in (33), a second regression model is introduced to estimate σ ²(t) (Wasserman 2006):

$$Z_{i}=\ln(Y_{i}-\hat{r}_{N}(\textbf{t}_{i}))^{2} $$

$$=\ln(\sigma^{2}(\textbf{t}_{i}){\kappa_{i}^{2}}) $$

$$ =\ln(\sigma^{2}(\textbf{t}_{i}))+\ln({\kappa_{i}^{2}}) $$

(40)

From (40) it can be observed that an estimate of ln(σ ²(t)) can be obtained by regressing Z _i ’s on t _i ’s. For example, a non-parametric regression model can be employed to obtain an estimate $\hat {\nu }(\textbf {t})$ of logσ ²(t). The estimate of the variance then becomes:

$$ \hat{\sigma}^{2}(\textbf{t})=\exp(\hat{\nu}(\textbf{t})) $$

(41)

A value of h for the kernel function in (37) is commonly selected by minimizing the leave-one-out cross-validation score (e.g., Wasserman2006):

$$ \hat{R}(h)=\frac{1}{N} \sum\limits_{i=1}^{N} \left( \frac{Y_{i}-\hat{r}_{N}(\textbf{t}_{i})}{1-\textbf{W}_{ii}} \right)^{2} $$

(42)

where W _{i i} = w _i (t _i ) is the ith diagonal element of the smoothing matrix.

Appendix: B: Analytical solution to the linear optimization problem

The RO problem in (24a–24d) can be solved analytically, based on the fact that g(u,t) is a linear combination of independent standard normally distributed random variables. Consequently, this leads to g(u,t) being a normally distributed random variable with mean, μ _g , and standard deviation, σ _g . The mean and the standard deviation are calculated to be, respectively, $\mu _{g}=\sum \limits _{i=1}^{n}t_{i}$ and $\sigma _{g}=\sqrt {m}$. The failure probability is calculated as:

$$ P_{F}(\textbf{t})= {\Phi} \left( -\frac{\mu_{g}}{\sigma_{g}} \right) = {\Phi} \left( -\frac{1}{\sqrt{m}} \sum\limits_{i=1}^{n}t_{i} \right)={\Phi} \left( -\beta(\textbf{t}) \right) $$

(43)

where Φ is the standard normal cumulative density function. With the analytical solution of the reliability problem in (43), the total cost is formulated as:

$$ C(\textbf{t})=\sum\limits_{i=1}^{n}C_{i} {t_{i}^{2}} +C_{F} {\Phi} \left( -\frac{1}{\sqrt{m}}\sum\limits_{i=1}^{n}t_{i} \right) $$

(44)

The minimum of the total cost is located by differentiating the cost function with respect to the design parameters, setting it equal to zero, and solving for the design parameters. The derivative of the cost function with respect to t _i is:

$$ \frac{\partial C(\textbf{t})}{\partial t_{i}}=2C_{i}t_{i}-\frac{C_{F}}{\sqrt{m}} \phi \left( -\frac{1}{\sqrt{m}} \sum\limits_{i=1}^{n}t_{i} \right) $$

(45)

where ϕ is the standard normal probability density function. After setting the derivative equal to zero, the following expression is obtained:

$$ t_{i}=\frac{C_{F}}{2C_{i}\sqrt{m}}\phi \left( -\frac{1}{\sqrt{m}}\sum\limits_{i=1}^{n}t_{i} \right) $$

(46)

Since (46) contains t _i on both sides, the ith component of the minimizer is defined by specifying a desired reliability index, β _min at the minimizer.

$$ \beta_{\min}=\frac{1}{\sqrt{m}}\sum\limits_{i=1}^{n}t_{i}=\frac{nt_{\min}}{\sqrt{m}} $$

(47)

where all the design parameters have the same value at the minimum, t _min. From (47) it follows:

$$ t_{\min}=\frac{1}{n} \beta_{\min} \sqrt{m} $$

(48)

In order for (46) to be consistent, the values of the design cost parameters, C _i , are defined based on the values of β _min, t _min, and C _F :

$$ C_{i}=\frac{C_{F}}{2t_{\min}\sqrt{m}}\phi(-\beta_{\min}) $$

(49)

An additional requirement for the results in (47) to (49) is that ${\Phi }(-\beta _{\min }) \le P_{F}^{\lim }$. Otherwise, the minimum is found at the reliability constraint.