# RBO using Global Response Surface

## General Remarks

Commonly, numerical efforts associated with the direct solution of a reliability-based optimization problem (RBO) problem are dominated by the reliability assessment step. Hence, one of the strategies that has been developed to alleviate the numerical costs when solving a RBO problem is constructing approximate representations of the reliability as an explicit function of the design variables, i.e. to decouple the optimization step from the reliability analysis. By means of such representation the RBO problem reduces to an ordinary non linear programming problem which can be solved by any appropriate optimization procedure.

Early efforts in the field of reliability analysis showed that rare occurrence events may be approximately represented by means of an exponential function [1][2]. Such an approach was employed in the context of RBO, e.g. in [3], in order to generate a global response surface of the failure probabilities as an explicit function of the design variables; the approximation was constructed by selecting some predefined interpolation points in the space of the design variables, at which the failure probability was calculated by means of simulation; then, an exponential function was adjusted to the data collected at the interpolation points in a least square sense.

## Example

The use of a global response surface for solving RBO problems is shown schematically in the figures below, where a two variable RBO problem is depicted ('DV' stands for design variable). Initially, some interpolation points (green dots in the figure below) are selected in the domain of the design variables. The probabilities are evaluated at these points.

Grid of points for constructing response surface of reliability

Once the probability has been evaluated, a Response Surface (RS) of the reliability is created.

Response surface approximating reliability function
Finally, the RBO problem is solved using the RS, using any appropriate optimization algorithm (each design tried by the optimization algorithm is denoted with the white dots in the figure below).
Optimization using response surface of reliability

## References

1. N.C. Lind. Approximate analysis and economics of structures. ASCE Journal of the Structural Division, 102, ST6:1177–1196, 1976.
2. J. Kanda and B. Ellingwood. Formulation of load factors based on optimum reliability. Structural Safety, 9(3):197–210, 1991.
3. M. Gasser and G.I. Schuëller. Reliability-based optimization of structural systems. Mathematical Methods of Operations Research, 46(3):287–307, 1997.[1]