| In engineering design, computer simulations are often used in optimization. To enhance robustness, it is necessary to include uncertainties that may come from physical randomness, insufficient knowledge or imperfect modeling. In this dissertation, the problem of optimization with noisy simulations is considered. The input parameters are separated into two groups: design variables and random factors. The optimization goal is to find the values for the design variables which minimize expected costs under uncertainties as quantified by the random factors. The approach is to integrate surrogate-surface methods into a framework of trust-region-based sequential minimization.; A significant portion of the dissertation is devoted to establishing provable convergence. Convergence proofs are derived for unconstrained and constrained optimization under a set of mathematical conditions for objective function uncertainty. The conditions are in terms of probabilistic bounds on the errors in the mean of the objective function and its gradient. If a Gaussian model is used for the errors, then the conditions can be simplified in terms of the bias, variance and mean-square values. These statistics can be estimated from simulation results.; To obtain a surrogate surface, which is simply an estimate of the mean of the objective function, local linear regression is used. This regression method is well suited for subsequent minimization by the trust-region algorithm because the support of the local regression kernel can be adapted to the size of the trust-region. Furthermore, the statistics of the errors in the regression fit can be derived in terms of a local second-order fit of the true mean function. The analysis of such a second-order fit is theoretically consistent with the second-order fit in the convergence proofs of the trust-region algorithm. Hence, it is possible to show in this dissertation, that an optimization approach based on local linear regression coupled with a trust-region algorithm is provably convergent.; To illustrate the surrogate-based optimization approach, a well-known test problem of truss design is analyzed. It is shown that optimization under uncertainty leads to a significantly different solution for the truss design as compared with an equivalent optimization problem under mean conditions. |
