Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint

¹ Laboratoire de l'Informatique Théorique et Appliquée, UFR Scientifique MIM Université Paul Verlaine – Metz, Ile du Saulcy, 57045 Metz, France; lethi@univ-metz.fr
² Département de Mathématiques, Faculté des Sciences, Université de Tizi-Ouzou, Algeria.

Received: 6 September 2005
Accepted: 6 April 2006

Abstract

The purpose of this paper is to demonstrate that, for globally minimize one dimensional nonconvex problems with both twice differentiable function and constraint, we can propose an efficient algorithm based on Branch and Bound techniques. The method is first displayed in the simple case with an interval constraint. The extension is displayed afterwards to the general case with an additional nonconvex twice differentiable constraint. A quadratic bounding function which is better than the well known linear underestimator is proposed while w-subdivision is added to support the branching procedure. Computational results on several and various types of functions show the efficiency of our algorithms and their superiority with respect to the existing methods.