## 4 Proposed Method for Sensitivity Analysis of SORM-Based Estimates
### Complex-Step Derivative Approximation  
The proposed method adopts the CS method [39] to perform the sensitivity analysis of SORM-based estimates associated with probabilistic constraints in the RBDO. [PERSON] and [PERSON] [40] proposed a method to compute the first derivative of a function by using the CS derivative approximation and demonstrated its higher efficiency and accuracy compared to those of the traditional finite difference methods [41]. Commonly, the first derivative in the finite difference method is estimated using a forward difference formula:  
\[f^{\prime}(a)\approx\frac{f(a+h)-f(a)}{h}+O(h) \tag{16}\]  
where \(h\) is the step size, and \(O(h)\) is the truncation error. A small step size, \(h\), is commonly selected to minimize the truncation error. However, an extremely small step size may cause a significant error due to dominant subtractive cancellation. The level of accuracy in the first derivative using the conventional finite difference method may vary with changes in the step size. The appropriate step size is generally unknown before actual calculations. Although higher-order finite difference methods can be adopted to minimize truncation errors, subtractive cancellation errors remain and may become significant in the calculations. In the CS derivative approximation, an imaginary number is used to develop an estimate of derivatives. Consider a differentiable function \(f(z)=u+\mathrm{i}v\) of the complex variable \(z=a+\mathrm{i}b\) (\(a,b\in\mathbb{R}\), \(\mathrm{i}^{2}=-1\)). Assume \(u\) and \(v\) are the real and imaginary parts \(f(z)\) respectively. Based on the Cauchy-Riemann equations that indicate the relationship between \(u\) and \(v\),  
\[ \begin{array}{c}\frac{\partial u}{\partial a}=\frac{\partial v}{\partial b} \\ \frac{\partial v}{\partial b}=-\frac{\partial v}{\partial a}\end{array}  \tag{17}\]  
The first Cauchy-Riemann equation yields:  
\[\frac{\partial u}{\partial a}=\lim_{h\to 0}\frac{v(a+\mathrm{i}(b+h))-v(a+ \mathrm{i}b)}{h},\;(h\in\mathbb{R}) \tag{18}\]  
By setting \(b=0\), \(z\) becomes a real number, such that \(f(a)=u(a)\) and \(v(a)=0\); then, the first-derivative expression in Equation (18) can be written as:  
\[\frac{\partial f(a)}{\partial a}=\lim_{h\to 0}\frac{\Im[f(a+\mathrm{i}h)]}{h},\;(h \in\mathbb{R}) \tag{19}\]  
For a small step size, \(h\), Equation (19) can be approximated as:  
\[\frac{\partial f(a)}{\partial a}\approx\frac{\Im[f(a+\mathrm{i}h)]}{h} \tag{20}\]where \(\Im[\cdot]\) denotes the imaginary part of the function \(f\). Evaluating \(f\) at the imaginary argument \(a+\mathrm{i}h\) and dividing it by \(h\) yields an approximation to the first derivative. Note that a truncation error, \(O(h^{2})\), that can be confirmed by the use of a Taylor series expansion at \(a+\mathrm{i}h\), exists in Equation (20). However, the CS derivative approximation does not include a subtraction operation, and thus, the calculation results are not affected by the subtractive cancellation and its associated round-off errors. Multiplying the real part of Equation (20) with \(h\) yields a value of function \(f(a)\). In other words, the first derivative and function value can be computed simultaneously during the implementation. It should be noted that sensitivity analysis using the CS derivative approximation simply requires the evaluation of the function with a complex variable, without analytical derivations of gradients. Therefore, the CS derivative approximation allows for sensitive analysis, even when the analytic derivation of gradients cannot be explicitly expressed, or each analysis becomes computationally expensive. Sensitivity Analysis of the Failure Probability Obtained Using the SORM via the CS Derivative Approximation  
Consider a vector of realization of \(n\) random variables \(\mathbf{x}=(x_{1}x_{2}x_{3},\,\ldots\,x_{n})^{\mathrm{T}}\) and a vector of \(z\) design variables \(\mathbf{d}=(d_{1}d_{2}d_{3},\,\ldots\,d_{z})^{\mathrm{T}}\).