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First- and Second-Order Asymptotic Analysis with Applications in Quasiconvex Optimization

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Abstract

We use asymptotic analysis to describe in a more systematic way the behavior at the infinity of functions in the convex and quasiconvex case. Starting from the formulae for the first- and second-order asymptotic function in the convex case, we introduce similar notions suitable for dealing with quasiconvex functions. Afterward, by using such notions, a class of quasiconvex vector mappings under which the image of a closed convex set is closed, is introduced; we characterize the nonemptiness and boundedness of the set of minimizers of any lsc quasiconvex function; finally, we also characterize boundedness from below, along lines, of any proper and lsc function.

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References

  1. Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer-Verlag, New York, Berlin (2003)

    MATH  Google Scholar 

  2. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  3. Hadjisavvas, N., Luc, D.T.: Second-order asymptotic directions of unbounded sets with application to optimization. J. Convex Anal. 18, 181–202 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Flores-Bazán, F., Hadjisavvas, N., Lara, F.: Second order asymptotic analysis: basic theory. J. Convex Anal. 22, 1173–1196 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Adly, S., Ernst, E., Théra, M.: Well-positioned closed convex sets and well-positioned closed convex functions. J. Global Optim. 29, 337–351 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. McFadden, D.: Convex analysis. In: Fuss, M., McFadden, D. (eds.) Production Economics: A Dual Approach to the Theory and Applications, vol. 1, pp. 383–408. North-Holland, Amsterdam (1979)

    Google Scholar 

  7. Amara, C.: Directions de majoration d’une fonction quasiconvexe et applications. Serdica Math. J. 24, 289–306 (1998)

    MathSciNet  Google Scholar 

  8. Penot, J.P.: Noncoercive problems and asymptotic conditions. Asymptot. Anal. 49, 205–215 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivates. J. Global Optim. 63, 99–123 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Penot, J.P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Penot, J.P.: Glimpses upon quasiconvex analysis. ESAIM: Proc. 20, 170–194 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Beer, G., Rockafellar, R.T., Wets, J.B.: A characterization of epi-convergence in terms of convergence of level sets. Proc. Am. Math. Soc. 16, 753–761 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Luc, D.T., Penot, J.P.: Convergence of asymptotic directions. Trans. Am. Math. Soc. 353, 4095–4121 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer-Verlag, New York, Berlin (1989)

    Google Scholar 

  15. Flores-Bazán, F.: Existence theory for finite-dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77, 249–297 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Komlósi, S.: Generalized convexity and generalized derivatives. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, pp. 421–463. Springer, Boston (2005)

    Chapter  Google Scholar 

  17. Deng, S.: Coercivity properties and well-posedness in vector optimization. RAIRO Oper. Res. 37, 195–208 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gwinner, J.: Closed images of convex multivalued mappings in linear topological spaces with applications. J. Math. Anal. Appl. 60, 75–86 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rockafellar, R.T., Wets, R.: Variational Analysis. Springer-Verlag, Berlin, Heidelberg (1998)

    Book  MATH  Google Scholar 

  20. Cotrina, J., Raupp, F.M.P., Sosa, W.: Semicontinuous quadratic optimization: existence conditions and duality scheme. J. Global Optim. 63, 281–295 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors want to express their gratitude to the referee for improvements he/she has suggested. This research, for the first author, was partially supported by CONICYT-Chile through FONDECYT 115-0973 and BASAL projects, CMM, Universidad de Chile; for the third author, his research was supported by CONICYT-Chile through Fondecyt PostDoctorado 3160205. Part of this work was carried out while the second author was visiting the Departamento de Ingeniería Matemática, Universidad de Concepción, Chile, from August 2013 to January 2014, and during July, 2014, supported in part by CONICYT through the Programa Atracción de Capital Humano Avanzado del Extranjero, MEC 2012, project 8012033, and FONDECYT 112-0980.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile

    F. Flores-Bazán & I. Montenegro

  2. Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran, Kingdom of Saudi Arabia

    N. Hadjisavvas

  3. Departamento de Matemáticas, Universidad de Tarapacá, Arica, Chile

    F. Lara

Authors
  1. F. Flores-Bazán
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  2. N. Hadjisavvas
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  3. F. Lara
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  4. I. Montenegro
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Corresponding author

Correspondence to F. Flores-Bazán.

Additional information

Communicated by Jonathan Michael Borwein.

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Flores-Bazán, F., Hadjisavvas, N., Lara, F. et al. First- and Second-Order Asymptotic Analysis with Applications in Quasiconvex Optimization. J Optim Theory Appl 170, 372–393 (2016). https://doi.org/10.1007/s10957-016-0938-6

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  • DOI: https://doi.org/10.1007/s10957-016-0938-6

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