Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-08T23:23:07.870Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally compactly Lipschitzian mappings in infinite dimensional programming

Published online by Cambridge University Press:  17 April 2009

B.M. Glover
B.M. Glover
Affiliation:
School of Mathematics and Computing, Ballarat University College, Ballarat Vic 3353, Australia
Rights & Permissions [Opens in a new window]

Extract

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 note we show that a subgradient multifunction of a locally compactly Lip-schitzian mapping satisfies a closure condition used extensively in optimisation theory. In addition we derive a chain rule applicable in either separable or reflexive Banach spaces for the class of locally compactly Lipschitzian mappings using a recently derived generalised Jacobian. We apply these results to the derivation of Karush-Kuhn-Tucker and Fritz John optimality conditions for general abstract cone-constrained programming problems. A discussion of constraint qualifications is undertaken in this setting.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 3 , June 1993 , pp. 395 - 406
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Borwein, J.M., ‘Stability and regular points of inequality systems’, J. Optim. Theory Appl. 48 (1986), 9–52.CrossRefGoogle Scholar
[2]Burke, J.V., ‘An exact penalization viewpoint of constrained optimization’, SIAM J. Control Optim. 29 (1991), 968–998.CrossRefGoogle Scholar
[3]Clarke, F.H., Optimization and nonsmooth analysis (Wiley Interscience, New York, 1983).Google Scholar
[4]Craven, B.D., Mathematical programming and control theory (Chapman-Hall, London, 1978).CrossRefGoogle Scholar
[5]Dien, P.H., ‘Locally Lipschitzian set-valued maps and generalized extremal problems with inclusion constraints’, Acta Math. Vietnam. 8 (1983), 109–122.Google Scholar
[6]Dien, P.H., ‘On the regularity condition for the extremal problem under locally Lipschitz inclusion constraints’, Appl. Math, and Optim. 13 (1985), 151–161.Google Scholar
[7]Dien, P.H. and Yen, N.D., ‘On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints’, Appl. Math. Optim. 24 (1991), 35–54.CrossRefGoogle Scholar
[8]Dien, P.H. and Sach, P.H., ‘Further properties of the regularity of inclusion systems’, Nonlinear Anal. 13 (1989), 1251–1267.CrossRefGoogle Scholar
[9]Ekeland, I., ‘On the variational principle’, J. Math. Anal. Appl. 47 (1974), 324–353.CrossRefGoogle Scholar
[10]Flåm, S., ‘Lagrangian conditions and stochastic programming’, SIAM J. Control Optim. 30 (1992), 1–10.CrossRefGoogle Scholar
[11]Glover, B.M., Jeyakumar, V. and Oettli, W., ‘A Farkas lemma for difference sublinear systems and quasidifferentiable programming’, School of Mathematics and Computing, Ballarat University College, Research Report 1/92, (submitted).Google Scholar
[12]Holmes, R.B., Geometric functional analysis and its applications (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[13]Ioffe, A.D., ‘Regular points of Lipschitz mappings’, Trans. Amer. Math. Soc. 251 (1979), 61–69.CrossRefGoogle Scholar
[14]Jourani, A. and Thibault, L., ‘Approximate subdifferential and metric regularity: The finite dimensional case’, Math. Programming 47 (1990), 203–218.CrossRefGoogle Scholar
[15]Lai, H.-C., ‘Optimality conditions for Lipschitz functions on Banach spaces’, Nihonkai Math. Journal 2 (1991), 99–115.Google Scholar
[16]Ralph, D., Rank-1 support functional and the generalized Jacobian, piecewise linear homeomorphisms, Ph.D Thesis (University of Wisconsin, Madison, 1990).Google Scholar
[17]Robinson, S.M., ‘Stability theory for systems of inequalities, Part II: Differentiate nonlinear systems’, SIAM J. Numer. Anal. 13 (1976), 497–513.CrossRefGoogle Scholar
[18]Sach, P.H., ‘Regularity, calmness and support principle’, Optimization 10 (1988), 13–27.CrossRefGoogle Scholar
[19]Thibault, L., ‘Subdifferentials of compactly Lipschitzian vector-valued functions’, Ann. Mat. Pura Appl. 125 (1980), 157–192.CrossRefGoogle Scholar
[20]Thibault, L., ‘On generalized differentials and subdifferentials of Lipschitz vector-valued functions’, Nonlinear Anal. 6 (1982), 1037–1053.CrossRefGoogle Scholar
[21]Thibault, L., ‘Subdifferentials of nonconvex vector-valued functions’, J. Math. Anal. Appl. 86 (1982), 319–354.CrossRefGoogle Scholar
[22]Yen, N.D., Stability of the solution set of perturbed nonsmooth inequality systems, International Centre for Theoretical Physics (International Atomic Energy Agency, UNESCO, Trieste, Italy, 1990).Google Scholar
[23]Yosida, K., Functional analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1980).Google Scholar
[24]Zowe, J. and Kurcyuz, S., ‘Regularity and stability of the mathematical programming problem in Banach spaces’, Appl. Math. Optim. 5 (1979), 49–62.Google Scholar