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Journal of Global Optimization

, Volume 71, Issue 4, pp 735–752 | Cite as

An edge-concave underestimator for the global optimization of twice-differentiable nonconvex problems

  • M. M. Faruque Hasan
Article

Abstract

We present a new relaxation method for the deterministic global optimization of general nonconvex and \({\mathscr {C}}^2\)-continuous problems. Instead of using a convex underestimator, the method uses an edge-concave (componentwise concave) underestimator to relax a nonconvex function. The underestimator is constructed by subtracting a positive quadratic expression such that all nonedge-concavities in the original function is overpowered by the added expression. While the edge-concave underestimator is nonlinear, the linear facets of its vertex polyhedral convex envelope leads to a linear programming (LP)-based relaxation of the original nonconvex problem. We present some theoretical results on this new class of underestimators and compare the performance of the LP relaxation with relaxations obtained by convex underestimators such as \(\alpha \hbox {BB}\) and its variants for several test problems. We also discuss the potential of a hybrid relaxation, relying on the dynamic selection of convex and edge-concave underestimators using criteria such as maximum separation distance.

Keywords

Global optimization Edge-concave underestimator Relaxation Nonconvex NLP 

Notes

Acknowledgements

Financial support from the U.S. National Science Foundation (Award Number CBET-1606027) is gratefully acknowledged. M.M.F.H. likes to thank Dr. Yannis Guzman and Dr. Eric First for their help in comparing results with previous methods.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Artie McFerrin Department of Chemical EngineeringTexas A&M UniversityCollege StationUSA

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