Economic inexact restoration for derivative-free expensive function minimization and applications

Abstract

The Inexact Restoration approach has proved to be an adequate tool for handling the problem of minimizing an expensive function within an arbitrary feasible set by using different degrees of precision in the objective function. The Inexact Restoration framework allows one to obtain suitable convergence and complexity results for an approach that rationally combines low-and high-precision evaluations. In the present research, it is recognized that many problems with expensive objective functions are nonsmooth and, sometimes, even discontinuous. Having this in mind, the Inexact Restoration approach is extended to the nonsmooth or discontinuous case. Although optimization phases that rely on smoothness cannot be used in this case, basic convergence and complexity results are recovered. A derivative-free optimization phase is defined and the subproblems that arise at this phase are solved using a regularization approach that take advantage of different notions of stationarity. The new methodology is applied to the problem of reproducing a controlled experiment that mimics the failure of a dam.

Inexact Restoration for Minimization with Inexact Evaluation both of the Objective Function and the Constraints

“…The papers [11,12,13,40] suggested that a good framework to address this problem is given by the Inexact Restoration approach of classical constrained optimization. The idea is that "maximal evaluation precision" can be considered as a constraint of the problem depending of a precision variable y that lies in an abstract set Y .…”

“…This basic principle was developed in [11] and [12], where complexity results were also proved. Moreover, in [13] the case in which derivatives are not available was considered. Inexactness of the objective function in optimization problems was addressed in several additional papers in recent years [7,8,9,34,35,39,33].…”

“…Define N reg = ⌊log 2 (C µ ) − log 2 (µ min )⌋ + 1 and μ = max{10C µ , 10 Nacce µ max }. Then, after at most N reg sub-iterations at Step 4 of BIRA, the conditions(13) and(14) are fulfilled . Moreover, µ k ≤ μ for all k.…”