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Transformation of Quasiconvex Functions to Eliminate Local Minima

  • Suliman Al-Homidan
  • Nicolas Hadjisavvas
  • Loai Shaalan
Article

Abstract

Quasiconvex functions present some difficulties in global optimization, because their graph contains “flat parts”; thus, a local minimum is not necessarily the global minimum. In this paper, we show that any lower semicontinuous quasiconvex function may be written as a composition of two functions, one of which is nondecreasing, and the other is quasiconvex with the property that every local minimum is global minimum. Thus, finding the global minimum of any lower semicontinuous quasiconvex function is equivalent to finding the minimum of a quasiconvex function, which has no local minima other than its global minimum. The construction of the decomposition is based on the notion of “adjusted sublevel set.” In particular, we study the structure of the class of sublevel sets, and the continuity properties of the sublevel set operator and its corresponding normal operator.

Keywords

Quasiconvex function Generalized convexity Adjusted sublevel sets Normal operator 

Mathematics Subject Classification

90C26 90C30 26A51 26B25 

Notes

Acknowledgements

The authors are grateful to KFUPM, Dhahran, Saudi Arabia, for providing excellent research facilities. They would also like to thank the referees for their useful comments and suggestions that helped to significantly improve the paper.

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Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of Product and Systems Design EngineeringUniversity of the AegeanHermoupolisGreece

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