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.

@article{RO_2006__40_3_285_0, author = {Le Thi, Hoai An and Ouanes, Mohand}, title = {Convex quadratic underestimation and {Branch} and {Bound} for univariate global optimization with one nonconvex constraint}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {285--302}, publisher = {EDP-Sciences}, volume = {40}, number = {3}, year = {2006}, doi = {10.1051/ro:2006024}, mrnumber = {2276160}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro:2006024/} }

TY - JOUR AU - Le Thi, Hoai An AU - Ouanes, Mohand TI - Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2006 SP - 285 EP - 302 VL - 40 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro:2006024/ DO - 10.1051/ro:2006024 LA - en ID - RO_2006__40_3_285_0 ER -

%0 Journal Article %A Le Thi, Hoai An %A Ouanes, Mohand %T Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint %J RAIRO - Operations Research - Recherche Opérationnelle %D 2006 %P 285-302 %V 40 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro:2006024/ %R 10.1051/ro:2006024 %G en %F RO_2006__40_3_285_0

Le Thi, Hoai An; Ouanes, Mohand. Convex quadratic underestimation and Branch and Bound for univariate global optimization with one nonconvex constraint. RAIRO - Operations Research - Recherche Opérationnelle, Volume 40 (2006) no. 3, pp. 285-302. doi : 10.1051/ro:2006024. http://archive.numdam.org/articles/10.1051/ro:2006024/

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