Approximating the probability density function of the optimal point of an optimization problem

This article aims at approximating the probability density function (PDF) of the optimal point of an optimization process. The full characterization of the PDF of the optimum is expected to lead to more precise failure prevision and increased safety with a cheaper design when compared with less accurate approaches such as those which approximate the random variables using only their mean and variance. The polynomial chaos expansion (PCE) is employed and the resulting functional is minimized using stochastic approximation techniques. Several non-convex functions and a laminated composite plate optimization problem are analysed and the validation of the proposed methodology is done comparing its results to those obtained using the Monte Carlo Simulation (MCS). The numerical analysis shows that the proposed methodology has successfully approximated the PDF of the solution of the optimization process of all the tested functions.