Managing the Choice of Surrogate Variables and the Use of Approximation Models to Optimize Expensive Functions

摘要

We propose research in three basic areas. First, it is standard in engineering practice to use approximation models of expensive simulations to drive nonlinear programming algorithms. An open question, which we will investigate using well-established notions from the literature on trust-region methods, is how to manage the interplay between optimization and the fidelity of the approximation models to insure that the process converges to a reasonable solution of the original design problem. It is also standard in engineering design to reduce the dimension of the optimization problem to be solved by using a technique known as variable linking. We plan to generalize this concept using some recent ideas from the optimization community to design robust ways to decompose a single iteration of any optimization method into a multi-phase process. First, an algorithm is applied to solve subproblems posed on subspaces of lower dimension, and then to solve the full problem on the affine hull defined by the current iterate and the solutions found for the subproblems. Crucial questions remain about the convergence of a practical implementation of this process. Finally, we will continue our research to extend and analyze pattern search methods. Pattern search methods can be successfully applied when only ranking (ordinal) information is available and when derivatives are either unavailable or unreliable. Since these are situations that occur in the design problems of interest, we propose to continue our investigation of pattern search methods. In particular, we will examine robust extensions to handle problems with bound constraints.