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On second-order conditions in unconstrained optimization

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Abstract

The main purpose of this paper is to establish the second-order nonsmooth sufficient unconstrained optimality condition for so called ℓ-stable at some point functions and in this way to generalize some previous results in this direction. We provide the comparisons with other results by examples.

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References

  1. Auslender A. (1984). Stability in mathematical programming with nondifferentiable data. SIAM J. Control Optim. 22: 239–254

    Article  MATH  MathSciNet  Google Scholar 

  2. Bednařík, D., Pastor, K.: A technical note concerning one example. Nonlinear Anal. 55, 187–189 (2003)

    Google Scholar 

  3. Bednařík D. and Pastor K. (2004). On characterization of convexity for regularly locally Lipschitz functions. Nonlinear Anal. 57: 85–97

    Article  MathSciNet  MATH  Google Scholar 

  4. Bednařík D. and Pastor K. (2004). Elimination of strict convergence in optimization. SIAM J. Control Optim. 43(3): 1063–1077

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Tal A. and Zowe J. (1982). Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Math. Program. 24: 70–91

    Article  MATH  MathSciNet  Google Scholar 

  6. Ben-Tal A. and Zowe J. (1985). Directional derivatives in nonsmooth optimization. J. Optim. Theory Appl. 47: 483–490

    Article  MATH  MathSciNet  Google Scholar 

  7. Chan W.L., Huang L.R. and Ng K.F. (1994). On generalized second-order derivatives and Taylor expansions in nonsmooth optimization. SIAM J. Control Optim. 32: 591–611

    Article  MATH  MathSciNet  Google Scholar 

  8. Chaney R.W. (1987). Second-order necessary conditions in constrained semismooth optimization. SIAM J. Control Optim. 25: 1072–1081

    Article  MATH  MathSciNet  Google Scholar 

  9. Chaney, R.W.: Second-order sufficient conditions in nonsmooth optimization. Math. Oper. Res. 13, 660–673 (1988)

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Georgiev P.G. and Zlateva N.P. (1996). Second-order subdifferentials of C 1,1 functions and optimality conditions. Set-Valued Anal. 4: 101–117

    Article  MATH  MathSciNet  Google Scholar 

  13. Ginchev I., Guerraggio A. and Rocca M. (2006). From scalar to vector optimization. Appl. Math. 51: 5–36

    Article  MathSciNet  MATH  Google Scholar 

  14. Hiriart–Urruty J.B., Strodiot J.J. and Nguyen V.H. (1984). Generalized Hessian matrix and second-order optimality conditions for problems with C 1,1 data. Appl. Math. Optim. 11: 43–56

    Article  MATH  MathSciNet  Google Scholar 

  15. Huang L.R. and Ng K.F. (1994). Second-order necessary and sufficient conditions in nonsmooth optimization. Math. Program. 66: 379–402

    Article  MathSciNet  Google Scholar 

  16. Huang L.R. and Ng K.F. (1996). On some relations between Chaney’s generalized second-order directional derivative and that of Ben-Tal and Zowe. SIAM J. Control Optim. 34: 1220–1234

    Article  MATH  MathSciNet  Google Scholar 

  17. Huang L.R. and Ng K.F. (1997). On lower bounds of the second-order directional derivatives of Ben-Tal-Zowe and Chaney. Math. Oper. Res. 22: 747–753

    MATH  MathSciNet  Google Scholar 

  18. Kawasaki H. (1988). An envelope-like effect of infinite many inequality constraints on second-order necessary conditions for minimization problems. Math. Program. 41: 73–96

    Article  MATH  MathSciNet  Google Scholar 

  19. Klatte D. (2000). Upper Lipschitz behavior of solutions to perturbed C 1,1 programs. Math. Program. (Ser B) 88: 285–311

    Article  MATH  MathSciNet  Google Scholar 

  20. Maruyama Y. (1990). Second-order necessary conditions for nonlinear optimization problems in Banach spaces and their application to an optimal control problem. Math. Oper. Res. 15: 467–482

    Article  MATH  MathSciNet  Google Scholar 

  21. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330, 331. Springer, Berlin (2006)

  22. Pastor K. (2001). On relations among the generalized second-order directional derivatives. Discuss. Math. Diff. Incl. Contr. Optim. 21: 235–247

    MATH  MathSciNet  Google Scholar 

  23. Pastor K. (2005). Convexity and generalized second-order derivatives for locally Lipschitz functions. Nonlinear Anal. 60: 547–555

    MATH  MathSciNet  Google Scholar 

  24. Peano G. (1891). Sulla formula di Taylor. Atti dell’ Accademia delle Science di Torino 27: 40–46

    Google Scholar 

  25. Qi L. (1994). Superlinearly convergent approximate Newton methods for LC 1 optimization problem. Math. Program. 64: 277–294

    Article  Google Scholar 

  26. Rockafellar R.T. (1989). Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. Math. Oper. Res. 14: 462–484

    MATH  MathSciNet  Google Scholar 

  27. Rockafellar, R.T., Wets, J.B.: Variational Analysis. Springer, New York (1998)

  28. Thompson, B.S.: Real Analysis. Springer, Berlin (1985)

  29. Torre, D.L., Rocca, M.: Remarks on second order generalized derivatives for differentiable functions with Lipschitzian jacobian. Appl. Math. E-Notes 3, 130–137 (2003)

    Google Scholar 

  30. Yang X.Q. (1996). On second-order directional derivatives. Nonlinear Anal. 26: 55–66

    Article  MATH  MathSciNet  Google Scholar 

  31. Yang X.Q. (1999). On relations and applications of generalized second-order directional derivatives. Nonlinear Anal. 36: 595–614

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Hradec Králové, Rokitanského 62, 500 03, Hradec Králové, Czech Republic

    Dušan Bednařík & Karel Pastor

Authors
  1. Dušan Bednařík
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  2. Karel Pastor
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Correspondence to Karel Pastor.

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Supported by the Council of Czech Government (MSM 6198959214).

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Bednařík, D., Pastor, K. On second-order conditions in unconstrained optimization. Math. Program. 113, 283–298 (2008). https://doi.org/10.1007/s10107-007-0094-8

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  • DOI: https://doi.org/10.1007/s10107-007-0094-8

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