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Transformation of Quasiconvex Functions to Eliminate Local Minima

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

References

  1. 1.
    Cambini, A., Martein, L.: Generalized Convexity and Optimization. Springer, New York (2009)zbMATHGoogle Scholar
  2. 2.
    Lucchetti, R., Milasi, M.: Semistrictly quasiconcave approximation and an application to general equilibrium theory. J. Math. Anal. Appl. 428, 445–456 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borde, J., Crouzeix, J.-P.: Continuity properties of the normal cone to the level sets of a quasiconvex function. J. Optim. Theory Appl. 66, 415–429 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aussel, D., Daniilidis, A.: Normal characterization of the main classes of quasiconvex functions. Set Valued Anal. 8, 219–236 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aussel, D., Hadjisavvas, N.: Adjusted sublevel sets, normal operator and quasiconvex programming. SIAM J. Optim. 16, 358–367 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aussel, D., Pistek, M.: Limiting normal operator in quasiconvex analysis. Set Valued Var. Anal. 23, 669–685 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Luc, D.T., Penot, J.-P.: Convergence of asymptotic directions. Trans. Am. Math. Soc. 353, 4095–4121 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, Vol. I: Theory. Kluwer Academic Publishers, Norwell (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Crouzeix, J.-P., Eberhard, A., Ralph, D.: A geometrical insight on pseudoconvexity and pseudomonotonicity. Math. Program. 123, 61–83 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Crouzeix, J.-P., Eberhard, A., Ralph, D.: (Convex) level sets integration. J. Optim. Theory Appl. 171, 865–886 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. SIAM, Philadelphia (2010)CrossRefzbMATHGoogle Scholar

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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