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Minimal approximate Hessians for continuously Gâteaux differentiable functions

Published online by Cambridge University Press:  17 February 2009

Hongxu Li  and
Falun Huang
Hongxu Li
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China; e-mail: Hoxuli@sohu.com.
Falun Huang
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China; e-mail: Hoxuli@sohu.com.
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Abstract

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In this paper, we investigate minimal (weak) approximate Hessians, and completely answer the open questions raised by V. Jeyakumar and X. Q. Yang. As applications, we first give a generalised Taylor's expansion in terms of a minimal weak approximate Hessian. Then we characterise the convexity of a continuously Gâteaux differentiable function. Finally some necessary and sufficient optimality conditions are presented.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 45 , Issue 3 , January 2004 , pp. 349 - 359
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Ben-Tal, A. and Zowe, J., “A unified theory of first and second order conditions for extremum problems in topological vector spaces”, Math. Programming Study 19 (1982) 39–76.CrossRefGoogle Scholar
[2]Burke, J. V., “Second-order necessary and sufficient conditions for convex composite NDO”, Math. Programming 38 (1987) 287–302.CrossRefGoogle Scholar
[3]Burke, J. V. and Poliquin, R. A., “Optimality conditions for nonfinite-valued convex composite functions”, Math. Programming Ser. B 57 (1992) 103–120.CrossRefGoogle Scholar
[4]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
[5]Chaney, R. W., “Second-order directional derivatives for nonsmooth functions”, J. Math. Anal. Appl. 128 (1987) 495–511.CrossRefGoogle Scholar
[6]Clarke, F. H., Optimization and nonsmooth analysis (Wiley-Interscience, New York, 1983).Google Scholar
[7]Cominetti, R. and Correa, R., “A generalized second-order derivative in nonsmooth optimization”, SIAM J. Control and Optim. 28 (1990) 789–809.CrossRefGoogle Scholar
[8]Correa, R., Jofre, A. and Thibault, L., “Characterization of lower semi-continuous convex functions”, Proc. Amer. Math. Soc. 116 (1992) 67–72.CrossRefGoogle Scholar
[9]Fiacco, A. V., “Second-order sufficient conditions for weak and strict constrained minima”, SIAM J. Appl. Math. 16 (1968) 105–108.CrossRefGoogle Scholar
[10]Fletcher, R., Practical methods of optimization 16 (John Wiley, New York, 1987).Google Scholar
[11]Hiriart-Urruty, J. B., Strodiot, J. J. and Hein Nguyen, V., “Generalized Hessian matrix and second-order optimality conditions for problems with C 1.1 data”, Appl. Math. Optim. 11 (1984) 43–56.CrossRefGoogle Scholar
[12]Ioffe, A. D., “Necessary and sufficient conditions for a local minimum. Part 3: Second-order conditions and augmented duality”, SIAM J. Control Optim. 17 (1979) 266–288.CrossRefGoogle Scholar
[13]Jeyakumar, V. and Yang, X. Q., “Convex composite minimization with C 1.1 functions”, J. Optim. Theory Appl. 86 (1995) 631–648.CrossRefGoogle Scholar
[14]Jeyakumar, V. and Yang, X. Q., “Approximate generalized Hessians and Taylor's expansions for continuously Gâteaux differentiable functions”, Non. Anal. T. M. A. 36 (1999) 353–368.CrossRefGoogle Scholar
[15]Kawasaki, H., “An envelope-like effect of infinitely many inequality constrains on second-order necessary conditions for minimization problems”, Math. Programming 41 (1988) 73–96.CrossRefGoogle Scholar
[16]Michel, P. and Penot, J.-P., “A generalized derivative for calm and stable functions”, Differential Integral Equations 5 (1992) 433–454.CrossRefGoogle Scholar
[17]Pales, Z. and Zeidan, V., “Generalized Hessian for C 1.1 functions in infinite dimensional normed spaces”, Math. Programming 74 (1996) 59–78.CrossRefGoogle Scholar
[18]Qi, L., “Superlinearly convergent approximate Newton methods for LC 1 optimization problems”, Math. Programming 64 (1994) 277–294.CrossRefGoogle Scholar
[19]Rockafellar, R. T., “First-and second-order epidifferentiability in nonlinear programming”, Trans. Amer Math. Soc. 307 (1988) 75–108.CrossRefGoogle Scholar
[20]Rockafellar, R. T., “Second-order optimality conditions in nonlinear programming obtained by way of eipi-derivatives”, Math. Oper. Res. 14 (1989) 462–484.CrossRefGoogle Scholar
[21]Yang, X. Q., “Generalized second-order characterizations of convex functions”, J. Optim. Theory Appl. 82 (1994) 173–180.CrossRefGoogle Scholar
[22]Yang, X. Q., “Second-order conditions in C 1.1 optimization with applications”, Numer. Fund. Anal. Optim. 14 (1993) 621–632.CrossRefGoogle Scholar
[23]Yang, X. Q. and Jeyakumar, V., “Generalized second-order directional derivatives and optimization with C 1.1functions”, Optimization 26 (1992) 165–185.CrossRefGoogle Scholar