Value at risk for confidence level quantifications in robust engineering optimization

We show how to introduce the Value at Risk (VaR) concept in optimization algorithms with emphasis in calculation complexity issues. To do so, we assume known the PDF of the uncertainties. Our aim is to quantify our confidence on the optimal solution at low complexity without sampling of the control space. The notion of over‐solving appears naturally where it becomes useless to solve accurately near an optimum when the variations in control parameters fall below the uncertainties. Examples show the behavior of this VaR‐based correction and link the approach to momentum‐based optimization where the mean and variance of a functional are considered. The approach is then applied to an inverse problem with fluids with uncertainties in the definition of the injection devices. It is shown that an optimization problem with an admissible solution in the control space in the deterministic case can lose its solution in the presence of uncertainties on the control parameters, which suggests that the control space itself should be redefined in such a situation to recover an admissible problem. This permits to evaluate the cost of making reliable a system that has been deterministically designed but has uncertain parameters. A shape optimization problem closes the paper showing the importance of including VaR information during the design iterations and not only at the end of the procedure.