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The second order optimality conditions for nonlinear mathematical programming with C 1,1 data

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Abstract

To find nonlinear minimization problems are considered and standard C 2-regularity assumptions on the criterion function and constrained functions are reduced to C 1,1-regularity. With the aid of the generalized second order directional derivative for C 1,1 real-valued functions, a new second order necessary optimality condition and a new second order sufficient optimality condition for these problems are derived.

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References

  1. M. Avriel: Nonlinear Programming: Analysis and Methods. Prentice-Hall, New Jersey, 1976.

    Google Scholar 

  2. M. S. Bazzaraa and C. M. Shetty: Foundations of Optimization. Springer-Verlag, Berlin, Heidelberg, New York, 1976, pp. 73–80.

    Google Scholar 

  3. A. Ben-Tal: Second order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31 (1980), 143–165.

    Google Scholar 

  4. A. Ben-Tal: Second order theory of extremum problems. in: Fiacoo, A. V. and Kortanek, K. eds., Extremal methods and system analysis, Springer, Berlin (1980), 336–356.

    Google Scholar 

  5. A. Ben-Tal and J. Zowe: A unified theory of first and second order conditions for extremum problems in topological vector space. Math. Programming Stud. 19 (1982), 39–76.

    Google Scholar 

  6. R. W. Chancy: Second order necessary conditions in constrained semismooth optimization. SIAM J. Control Optim. 25 (1987), 1072–1081.

    Google Scholar 

  7. F. H. Clarke: Optimization and Nonsmooth Analysis. John Wiley and Sons, New York, 1983.

    Google Scholar 

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

    Google Scholar 

  9. K. H. Hoffmann and H. F. Kornstaedt: Higher order necessary conditions in abstract mathematical programming. J. Optim. Theory Appl. 26 (1978), 533–569.

    Google Scholar 

  10. A. D. Ioffe: Necessary and sufficient conditions for a local minimum 3: Second order conditions and augmented duality. SIAM J. Control Optim. 17 (1979), 266–288.

    Google Scholar 

  11. M. Křížek and P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Longman, 1990.

  12. J. Liu and B-G. Liu: On second order sufficient conditions in smooth nonlinear programming. In: Proc. Conf. Recent trends in optimization theory and applications, World Sci. Publ. Comp., 1995, 239–254.

  13. L. Liu: The second-order conditions of nondominated solutions for C 1,1 generalized multiobjective mathematical programming. J. Systems Sci. Math. Sci. 4 (1991), 128–138.

    Google Scholar 

  14. L. Liu: The second order conditions for C 1,1 nonlinear mathematical programming. Proc. Prague Math. Conf. 96, Math. Inst., Acad. Sci., Prague, 1996, 153–158.

  15. J. J. Maurer and J. Zowe: First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Programming 16 (1979), 98–110.

    Google Scholar 

  16. G. P. McCormick: Second order conditions for constrained minima. SIAM. J. Appl. Math. 15 (1967), 641–652.

    Google Scholar 

  17. S. Saks: Theory of the Integral. Hafner Publishing Co., New York, 1937.

    Google Scholar 

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Liu, L., Křížek, M. The second order optimality conditions for nonlinear mathematical programming with C 1,1 data. Applications of Mathematics 42, 311–320 (1997). https://doi.org/10.1023/A:1023068513188

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