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.
Similar content being viewed by others
References
M. Avriel: Nonlinear Programming: Analysis and Methods. Prentice-Hall, New Jersey, 1976.
M. S. Bazzaraa and C. M. Shetty: Foundations of Optimization. Springer-Verlag, Berlin, Heidelberg, New York, 1976, pp. 73–80.
A. Ben-Tal: Second order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31 (1980), 143–165.
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.
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.
R. W. Chancy: Second order necessary conditions in constrained semismooth optimization. SIAM J. Control Optim. 25 (1987), 1072–1081.
F. H. Clarke: Optimization and Nonsmooth Analysis. John Wiley and Sons, New York, 1983.
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.
K. H. Hoffmann and H. F. Kornstaedt: Higher order necessary conditions in abstract mathematical programming. J. Optim. Theory Appl. 26 (1978), 533–569.
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.
M. Křížek and P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Longman, 1990.
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.
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.
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.
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.
G. P. McCormick: Second order conditions for constrained minima. SIAM. J. Appl. Math. 15 (1967), 641–652.
S. Saks: Theory of the Integral. Hafner Publishing Co., New York, 1937.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1023/A:1023068513188