On the Global Convergence of Trust Region Algorithms Using Inexact Gradient Information

Abstract

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Analysis of Inexact Trust-Region SQP Algorithms

“…In the absence of inexactness our global convergence theory is that of [10]. If all iterates are feasible, i.e., if all iterates satisfy C(y k u k ) = 0 , then our results are related to the convergence analyses in [3,5] for trust-region methods with inexact function and gradient information for unconstrained optimization.…”

“…In section 2 we will consider the reduced problem min f(y(u) u ) obtained from (1.1) by eliminating the variables y. We will briefly discuss the convergence analyses in [3] and [5, xx 8.4,10.6] for trust-region methods with inexact function or gradient information for the reduced problem. This will reveal some useful problem information and it will later motivate our assumptions on the inexactness for problem (1.1).…”

A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization

“…It has been shown, however, that trust region optimizations converge even if inexact model and gradient information is used [7,15,28]. In [33], Yue and Meerbergen relax the stringent firstorder accuracy requirements to consider the general setting of an unconstrained trust region optimization algorithm using surrogate models with the following properties:…”

“…These local approximations automatically satisfy first-order consistency conditions (i.e., the approximate model's objective and gradient evaluations are locally exact), which in turn provide guarantees that the resulting optimization solution will satisfy the optimality conditions of the original high-fidelity system. The influence of inexact gradient information is considered in [7,28], and of inexact gradient and function information in [6,8,9]. In [1], the authors consider a trust region framework with more general approximation models of varying fidelity and show how adaptive corrections may be used to achieve the first-order consistency conditions required to achieve a provably convergent formulation for general approximation models.…”

A Provably Convergent Multifidelity Optimization Algorithm not Requiring High-Fidelity Derivatives

“…The challenge we address here is twofold, first we must produce a surrogate model that captures local function behavior sufficiently well to prove convergence without requiring a highfidelity gradient estimate, and secondly we must ensure the surrogate captures some global function behavior to speed convergence to a stationary point of the high-fidelity function. A trust region algorithm is convergent provided either the error between the gradient of the function and the gradient of surrogate model is bounded by a constant times the gradient of the function [21] or provided the accuracy of the surrogate can be improved dynamically within the trust region framework [ [22], section 10.6]. Oeuvray [23] showed that a radial basis function interpolation satisfies these criteria, provided the interpolation points satisfy certain conditions.…”