Use of quadratic models with mesh-adaptive direct search for constrained black box optimization

Abstract

We consider a derivative-free optimization, and in particular black box optimization, where the functions to be minimized and the functions representing the constraints are given by black boxes without derivatives. Two fundamental families of methods are available: model-based methods and directional direct search algorithms. This work exploits the flexibility of the second type of methods in order to integrate to a limited extent the models used in the first family. Intensive numerical tests on two sets of 48 and 104 test problems illustrate the efficiency of this hybridization and show that the use of the models improves the performance of the mesh- adaptive direct search algorithm significantly.

Keywords:

• black box optimization
• constrained optimization
• quadratic models
• mesh-adaptive direct search (MADS) algorithms

Acknowledgements

We thank Ana Custódio and Stefan Wild for having made their test problems and results available. We also thank Charles Audet, Ana again, John Dennis, Luís Vicente, and two anonymous referees for their useful comments.

Notes

It may be more natural to be concerned about WP in the undetermined case, but since we did not want to incur the costs of new function evaluations and in addition we assumed that the direct search method promotes good geometry naturally, we only considered the overdetermined case.

That is, oriented with the coordinate axes.