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

TL;DR

This paper extends the Inexact Restoration method to handle nonsmooth and discontinuous expensive functions using derivative-free optimization, demonstrating its application to a dam failure simulation.

Contribution

It introduces a novel extension of the Inexact Restoration framework for nonsmooth and discontinuous problems with derivative-free methods.

Findings

Convergence and complexity results are established for the extended method.

The approach is successfully applied to a dam failure simulation.

The method effectively handles nonsmooth, discontinuous, and expensive functions.

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.