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Generalized subdifferentials of the sign change counting function

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Abstract

The counting function on binary values is extended to the signed case in order to count the number of transitions between contiguous locations. A generalized subdifferential for this sign change counting function is given where classical subdifferentials remain intractable. An attempt to prove global optimality at some point, for the 4-dimensional first non trivial example, is made by using a sufficient condition specially tailored among all the cases for this subdifferential.

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References

  1. Carrasco-Olivera, D., Flores-Bazán, F.: On the representation of approximate subdifferentials for a class of generalized convex functions. Set Valued Anal. 13(2), 151–166 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Elkadi, M., Mourrain, B.: Some Applications of Bezoutians in Effective Algebraic Geometry. Technical Report RR-3572, INRIA, Dec 1998

  3. Flores-Bazán, F.: On minima of the difference of functions. J. Optim. Theory Appl. 93(3), 525–531 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fortin, D., Tseveendorj, I.: \(Q\)-subdifferential and \(Q\)-conjugate for global optimality. Comput. Math. Math. Phys. 54(2), 265–274 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hiriart-Urruty, J.-B., Lewis, A.S.: The Clarke and Michel-Penot subdifferentials of the eigenvalues of a symmetric matrix. Comput. Optim. Appl. 13(1–3), 13–23 (1999). Computational optimization—a tribute to Olvi Mangasarian, Part II

    Article  MathSciNet  MATH  Google Scholar 

  6. Hiriart-Urruty, J.-B., Le, H.Y.: Convexifying the set of matrices of bounded rank: applications to the quasiconvexification and convexification of the rank function. Optim. Lett. 6(5), 841–849 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. 110(3, Ser. A), 521–541 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Le, H.Y.: Generalized subdifferentials of the rank function. Optim. Lett. 7(4), 731–743 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lewis, A.S.: Nonsmooth analysis of eigenvalues. Math. Program. 84(1, Ser. A), 1–24 (1999)

    MathSciNet  MATH  Google Scholar 

  10. Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. I. Theory. Set Valued Anal. 13(3), 213–241 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. II. Applications. Set Valued Anal. 13(3), 243–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rubinov, A.M., Wu, Z.Y.: Optimality conditions in global optimization and their applications. Math. Program. 120(1, Ser. B), 101–123 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rubinov, A.: Abstract Convexity and Global Optimization, Volume of 44 Nonconvex Optimization and Its Applications. Kluwer, Dordrecht (2000)

    Book  Google Scholar 

  14. Tsevendorj, I.: Piecewise-convex maximization problems: global optimality conditions. J. Glob. Optim. 21(1), 1–14 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wu, Z.Y., Jeyakumar, V., Rubinov, A.M.: Sufficient conditions for global optimality of bivalent nonconvex quadratic programs with inequality constraints. J. Optim. Theory Appl. 133(1), 123–130 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wu, Z.Y., Rubinov, A.M.: Global optimality conditions for some classes of optimization problems. J. Optim. Theory Appl. 145(1), 164–185 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Inria, Domaine de Voluceau, Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Dominique Fortin

  2. Laboratoire PRiSM, Université de Versailles, 45, avenue des États-Unis, 78035, Versailles Cedex, France

    Ider Tseveendorj

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  1. Dominique Fortin
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  2. Ider Tseveendorj
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Correspondence to Dominique Fortin.

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Fortin, D., Tseveendorj, I. Generalized subdifferentials of the sign change counting function. J Glob Optim 65, 41–56 (2016). https://doi.org/10.1007/s10898-015-0332-1

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  • DOI: https://doi.org/10.1007/s10898-015-0332-1

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