Space Mapping: Models, Sensitivities, and Trust-Regions Methods

The goal of this paper is to organize some of the mathematical and algorithmic aspects of the space-mapping technique for continuous optimization with expensive function evaluations. First, we consider the mapping from the fine space to the coarse space when the models are vector-valued functions and when the space-mapping (nonlinear) least-squares residual is nonzero. We show how the sensitivities of the space mapping can be used to deal with space-mapping surrogates of the fine model. We derive a framework where it is possible to design globally convergent trust-region methods to minimize such fine-model surrogates. We consider also a different perspective of space mapping and apply it, for sake of simplicity, to the situation where the models are scalar functions. The space mapping is defined in a way where it is reasonable to assume that it is point-to-point. We prove that the surrogate model built by composition of the space mapping and the coarse model is a regular function. We also discuss trust-region methods in this context.