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Radiant Separation Theorems and Minimum-Type Subdifferentials of Calm Functions

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Abstract

Applying minimum-type functions and plus-cogauges, we construct a closed, convex cone in order to separate a boundary point of a radiant set from its interior. Abstract convexity of positively homogeneous functions is studied as well. Since a locally Lipschitz function is degree-one calm, the class of degree-one calm functions is large. We study degree-one calm functions and investigate how these functions can be generated by a class of min-type functions. Then, we derive a method to find an element of the subdifferential of a non-negative, lower semicontinuous and degree-one calm function with respect to the class of min-type functions.

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References

  1. Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Tammer, C.: A generalization of Ekeland’s variational principle. Optimization 25, 129–141 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Financ. 9, 203–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  6. Doagooei, A.R., Mohebi, H.: Monotonic analysis over ordered topological vector spaces: IV. J. Glob. Optim. 45, 355–369 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Doagooei, A.R.: Sub-topical functions and plus-co-radiant sets. Optimization 65, 107–119 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dutta, J., Martinez-Legaz, J.E., Rubinov, A.M.: Monotonic analysis over cones: I. Optimization 53, 165–177 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rubinov, A.M., Shveidel, A.P.: Radiant and star-shaped functions. Pac. J. Optim. 3, 193–212 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Nedic, A., Ozdaglar, A., Rubinov, A.M.: Abstract convexity for non-convex optimization duality. Optimization 56, 655–674 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Andramonov, M.Y., Rubinov, A.M., Glover, B.M.: Cutting angle method for minimizing convex along-rays functions. Research report 7/97. University of Ballarat, SITMS (1997)

  12. Andramonov, MYu., Rubinov, A.M., Glover, B.M.: Cutting angle methods in global optimization. Appl. Math. Lett. 12, 95–100 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rubinov, A.M., Vladimirov, A.: Convex along-rays functions and star-shaped sets. Numer. Funct. Anal. Optim. 19, 593–614 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rubinov, A.M., Sharikov, E.V.: Subdifferentials of convex along-rays functions. Optimization 56, 61–72 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Doagooei, A.R.: Minimum type functions, plus-cogauges and applications. J. Optim. Theory Appl. 164, 551–564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zaffaroni, A.: Is every radiant function the sum of quasiconvex functions? Math. Oper. Res. 59, 221–233 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zaffaroni, A.: Superlinear separation and dual properties of radiant functions. Pac. J. Optim. 2, 171–192 (2006)

    MATH  Google Scholar 

  18. Zaffaroni, A.: Superlinear separation for radiant and coradiant sets. Optimization 56, 267–285 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bagirov, A.M., Ugon, J., Webb, D.: An efficient algorithm for the incremental construction of a piecewise linear classifier. Inf. Syst. 36, 782–790 (2011)

    Article  Google Scholar 

  20. Bagirov, A.M., Ugon, J., Webb, D., Ozturk, G., Kasimbeyli, R.: A novel piecewise linear classifier based on polyhedral conic and max–min separabilities. Top 21, 3–24 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Martinez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. J. Optim. 19, 603–652 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Urfat, G.N.: An approach to the subproblem of the cutting angle method of global optimization. J. Glob. Optim. 31, 353–370 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Andramonov, M.Y.: Convergence of a numerical abstract convexity algorithm. Izvest. Math. 73, 3–19 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are very grateful to the anonymous referee, whose valuable comments and detailed remarks improved the presentation of the paper significantly.

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Authors and Affiliations

  1. Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

    A. Sheykhi

  2. Department of Applied Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

    A. R. Doagooei

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  1. A. Sheykhi
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  2. A. R. Doagooei
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Correspondence to A. R. Doagooei.

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Sheykhi, A., Doagooei, A.R. Radiant Separation Theorems and Minimum-Type Subdifferentials of Calm Functions. J Optim Theory Appl 174, 693–711 (2017). https://doi.org/10.1007/s10957-017-1132-1

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  • DOI: https://doi.org/10.1007/s10957-017-1132-1

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