# RBO using Local Response Surface

## General Remarks

A possible means for reducing the numerical efforts associated with the solution of RBO problems is the introduction of approximation concepts. In particular, the concept of response surface of the structural reliability can be applied. Such an idea has been used a number of times in the literature of RBO, see e.g. [1][2]. The advantage of this approach is that the reliability assessment step is decoupled from the optimization problem, i.e. the response surface is inexpensive to evaluate and thus, any appropriate algorithm can be used to solve the optimization problem. As the construction of this approximation over the entire domain of the design variables can be demanding, it may be easier to generate an approximation of the failure probabilities over a local domain [3], i.e. to generate a local response surface.

## Example

For a better understanding of the approach for solving RBO problems involving a local response surface and sequential approximations, consider the following RBO problem.

\begin{align} &\mbox{min }F(\mathbf{x}),~~~\mathbf{x}\in \Omega_x,~\Omega_x\subset\R^2 \\ &\mbox{subject to}\\ &p_{F}(\mathbf{x})\leq p_{F}^{tol} \end{align}

In the equation above, $\mathbf{x}$ is the vector of design variables, $F(\cdot)$ is an objective function, $p_F(\cdot)$ is a probability of failure and $p_F^{tol}$ is the maximum tolerable failure probability. A schematic representation of the solution of the optimization problem using local response surfaces and sequential approximations is shown in the figure below.

Optimization using local response surfaces of reliability

In the figure above, the continuous line indicates a contour level of the probability function; the segmented line, contour levels of the objective function and the dotted line, subdomains associated with each subproblem. The black dots denote candidate optimal designs. As depicted in the figure, the optimization procedure starts from an unfeasible design $\mathbf{x}^1$; around this design, a subdomain is defined and a local response surface of the reliability is constructed. Optimization in the subdomain is carried out and a new candidate optimal design, denoted as $\mathbf{x}^2$, is found. The whole process is repeated a number of times until finding the optimum solution at the fifth iteration.

A possible stopping criterion for the algorithm based on sequential approximations is, e.g. that the change in the value of the objective function between two successive iterations is below a prescribed threshold. In this context, it must be remarked that this optimization scheme does not guarantee that the global optimum can be determined. In fact, different starting points ($\mathbf{x}^1$) may lead to different local optima. However, this property does not impose a serious limitation, as usually engineering criteria and the knowledge on the problem at hand provide guidelines for assessing the quality of the approximate optimum determined using sequential approximations.