Error Analysis of Surrogate Models Constructed through Operations on Sub-models

Abstract

Model-based methods are popular in derivative-free optimization (DFO). In most of them, a single model function is built to approximate the objective function. This is generally based on the assumption that the objective function is one blackbox. However, some real-life and theoretical problems show that the objective function may consist of several blackboxes. In those problems, the information provided by each blackbox may not be equal. In this situation, one could build multiple sub-models that are then combined to become a final model. In this paper, we analyze the relation between the accuracy of those sub-models and the model constructed through their operations. We develop a broad framework that can be used as a theoretical tool in model error analysis and future research in DFO algorithms design.

Structure-Aware Methods for Expensive Derivative-Free Nonsmooth Composite Optimization

“…In each iteration of MS-P we first construct G k according to (3) and ensure that for each j ∈ G k , f j (x k ) + (g k j ) s is a gradient-accurate model of f j on B(x k ; ∆ k ). We remark that, given a method to construct fully linear (and, therefore, gradient-accurate) models M of F , it is straightforward to show under Assumption 4 that h(M (x)) is a gradient-accurate model of h(F (x)); see [18,Theorem 32]. We then (approximately 4 ) solve the subproblem (6) to obtain (v k , s k ).…”