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Approximate Hessian matrices and second-order optimality conditions for nonlinear programming problems with C1-data

Published online by Cambridge University Press:  17 February 2009

V. Jeyakumar  and
X. Wang
V. Jeyakumar
Affiliation:
Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia
X. Wang
Affiliation:
Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia
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Abstract

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In this paper, we present generalizations of the Jacobian matrix and the Hessian matrix to continuous maps and continuously differentiable functions respectively. We then establish second-order optimality conditions for mathematical programming problems with continuously differentiable functions. The results also sharpen the corresponding results for problems involving C1.1-functions.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 3 , January 1999 , pp. 403 - 420
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Chan, W. L., Huang, L. R. and Ng, K. F., ‘On generalized second-order derivatives and Taylor expansions in nonsmooth optimization’, SIAM J. Control Optim. 32 (1994) 591–611.CrossRefGoogle Scholar
[2]Clarke, F. H., Optimization and nonsmooth analysis (Wiley-Interscience, New York, 1983).Google Scholar
[3]Fiacco, A. V. and McCormick, G. P., Nonlinear Programming: Sequential unconstrained minimization techniques (SIAM Publication, Philadelphia, USA, 1990).CrossRefGoogle Scholar
[4]Hiriart-Urruty, J. B., ‘Mean value theorems for vector valued mappings in nonsmooth optimization’, Numer. Fund. Anal. Optim. 2 (1980) 1–30.CrossRefGoogle Scholar
[5]Hiriart-Urruty, J. B., Strodiot, J. J. and Nguyen, V. Hien, ‘Generalized Hessian matrix and secondorder optimality conditions for problems with C 1.1 data’, Applied Math. Optimiz. 11 (1984) 43–56.CrossRefGoogle Scholar
[6]Ioffe, A. D., ‘Nonsmooth analysis: differential calculus of nondifferential mappings’, Trans. Amer. Math. Soc. 266 (1981) 1–56.CrossRefGoogle Scholar
[7]Ioffe, A. D., ‘Approximate subdifferentials and applications I: The finite dimensional theory’, Trans. Amer. Math. Soc. 281 (1984) 389–416.Google Scholar
[8]Jeyakumar, V. and Luc, D. T., ‘Approximate Jacobian matrices for nonsmooth continuous maps and C 1-optimization’, SIAM J. Control and Optim. 36 (5) (1998) 1815–1832.CrossRefGoogle Scholar
[9]Jeyakumar, V., Luc, D. T. and Schaible, S., ‘Characterizations of generalized monotone nonsmooth continuous maps using approximate jacobians’, Applied Mathematics Research Report AMR 97/2, (University of New South Wales, Australia, 1997), J. Convex Analysis, to appear.Google Scholar
[10]Jeyakumar, V. and Yang, X. Q., ‘Approximate generalized Hessians and Taylor's expansions for continuously Gateaux differentiable functions’, Applied Mathematics Research Report AMR96/20, (University of New South Wales, Australia, 1996), Nonlinear Analysis T, M & A, to appear.Google Scholar
[11]Mordukhovich, B. S., ‘Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems’, Soviet. Math. Dokl. 22 (1980) 526–530.Google Scholar
[12]Mordukhovich, B. S., ‘Generalized differential calculus for nonsmooth and set-valued mappings’, J. Math. Anal. Appl. 183 (1994) 250–288.CrossRefGoogle Scholar
[13]Mordukhovich, B. S. and Shao, Y., ‘On nonconvex subdifferential calculus in banach spaces’, J. Convex. Anal. 2 (1995) 211–228.Google Scholar
[14]Rockafellar, R. T. and Wets, J. B., ‘Variational analysis’, (Springer Verlag, 1997).Google Scholar
[15]Studniarski, M. and Jeyakumar, V., ‘A generalized mean-value theorem and optimality conditions in composite nonsmooth minimization’, Nonlinear Anal. T.M.& A 24 (6) (1995) 883–894.CrossRefGoogle Scholar
[16]Yang, X. Q., ‘Second-order conditions for C 1.1 optimization with applications’, Numer. Funct. Anal. Optim. 14 (1993) 621–632.CrossRefGoogle Scholar
[17]Yang, X. Q. and Jeyakumar, V., ‘Generalized second-order directional derivatives and optimization with C 1.1 functions’, Optimization 26 (1992) 165–185.CrossRefGoogle Scholar