Title: Line-Search Based Optimization Using Function Approximations with Tunable Accuracy
Abstract: In application, many optimization problems require the solution of complex systems with inherent uncertainty or extreme complexity. In many cases, direct evaluation of these systems is infeasible or impossible and inaccurate approximations must be used. To address this problem, a line-search algorithm that uses objective function models with tunable accuracy to solve smooth optimization problems with convex constraints has been developed. This algorithm specifies how objective function models can be used to generate new iterates in the context of line-search methods, and specifies approximation properties these models have to satisfy. Moreover, the algorithm assumes that a bound for the model error is available and uses this bound to explore regions where the model is sufficiently accurate. The algorithm has the same first-order global convergence properties as standard line-search methods. However, this algorithm uses only the models and the model error bounds, but never directly accesses the original objective function. Examples include problems where the evaluation of the objective requires the solution of a large-scale system of nonlinear equations. The models are constructed from reduced order models of this system. Numerical results for partial differential equation constrained optimization problems show the benefits of the proposed algorithm.