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Constructing a sequence of discrete Hessian matrices of an SC 1 function uniformly convergent to the generalized Hessian matrix

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Abstract

We construct a uniform approximation for generalized Hessian matrix of an SC 1 function. Using the discrete gradient and the extended second order derivative, we define the discrete Hessian matrix. We construct a sequence of sets, where each set is composed of discrete Hessian matrices. We first show some new properties of SC 1 functions. Then, we prove that for SC 1 functions the sequence of the set of discrete Hessian matrices is uniformly convergent to the generalized Hessian matrix.

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References

  1. Bagirov A.: Continuous subdifferential approximations and their applications. J. Math. Sci. 115, 2567–2609 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ben-Tal A., Zowe J.: Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Math. Program. 24, 70–91 (1984)

    Article  MathSciNet  Google Scholar 

  3. Ioffe A.: Nonsmooth analysis: differential calculus of nondifferentiable mapping. Trans. Am. Math. Soc. 266, 1–55 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ioffe A., Milosz T.: On the characterization of C 1,1 functions. Cybern. Syst. Anal. 38, 313–322 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Rauf A.I., Fukushima M.: A globally convergent BFGS method for nonsmooth convex optimization. J. Optim. Theory Appl. 104, 539–558 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mordukhovich B.Sh.: Approximation Methods in Problems of Optimization and Control. Nauk, Moscow (1988)

    MATH  Google Scholar 

  7. Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  8. Clarke F.H., Ledyaev Yu., Stern R.J., Wolenski P.R.: Nonsmooth Analysis and Control Theory. Grad. Texts in Mathematics, vol 178. Springer, New York (1998)

    Google Scholar 

  9. Facchinei F.: Minimization of SC 1 functions and the Maratos effect. Oper. Res. Lett. 17, 131–137 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Xu H., Chang X.: Approximate Newton methods for nonsmooth equations. J. Optim. Theory Appl. 93, 373–394 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Xu H., Rubinov A., Glover B.M.: Continuous approximations to generalized Hessians. Optimization 46, 221–246 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Xu H.: Set-valued approximation and Newton’s methods. Math. Program. 84, 401–420 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hiriart-Urruty, J.-B.: The approximate second-order and second-order directional derivatives for a convex function. In: Cecconi, J.-P., Zolezzi, T. (eds.) Mathematical Theories of Optimization. Lecture Notes in Mathematics, vol. 979, pp. 144–177 (1983)

  14. Pang J.S., Qi L.: A globally convergence Newton method for SC 1 minimization problem. J. Optim. Theory Appl. 85, 635–648 (1995)

    Article  MathSciNet  Google Scholar 

  15. Pang J.S., Sun D., Sun J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28, 39–63 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Qi L., Sun J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–368 (1993)

    Article  MathSciNet  Google Scholar 

  17. Cominetti R., Correa R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control. Optim. 28, 789–809 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mifflin R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zeidan V.: Sufficient conditions for the generalized problem of Bolza. TAMS 275, 561–583 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  20. Chan W.L., Hung L.R., Ng K.F.: On Generalized second-order derivatives and Taylor expansion in nonsmooth optimization. SIAM J. Control Optim. 32, 591–611 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pales Z., Zeidan V.: Generalized Hessian for C 1,1 functions in infinite dimensional normed spaces. Math. Program. 74, 59–78 (1996)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Nezam Mahdavi-Amiri & Rohollah Yousefpour

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  1. Nezam Mahdavi-Amiri
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  2. Rohollah Yousefpour
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Correspondence to Nezam Mahdavi-Amiri.

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Mahdavi-Amiri, N., Yousefpour, R. Constructing a sequence of discrete Hessian matrices of an SC 1 function uniformly convergent to the generalized Hessian matrix. Math. Program. 121, 387–414 (2010). https://doi.org/10.1007/s10107-008-0238-5

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  • DOI: https://doi.org/10.1007/s10107-008-0238-5

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