Hostname: page-component-77f85d65b8-g98kq Total loading time: 0 Render date: 2026-04-18T16:13:32.022Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

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