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On metric regularity of multifunctions

Published online by Cambridge University Press:  17 April 2009

A. Jourani
A. Jourani
Affiliation:
Mathématiques, Université de Perpignan, 66025 Perpignan Cedex, France
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Abstract

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The aim of this paper is to give a metric regularity theorem for multifunctions between metric spaces involving some known results for multifunctions by using the notion of strict (G, δ)-differentiability of multifunctions and a simple convergence procedure.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 1 , August 1991 , pp. 1 - 9
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Aubin, J.P., ‘Lipschitz behavior of solutions to convex minimization problems’, Math. Oper. Res. 9 (1984), 87–111.CrossRefGoogle Scholar
[2]Aubin, J.P. and Frankowska, H., ‘On the inverse function theorem’, J. Math. Pures Appl. 66 (1987), 71–89.Google Scholar
[3]Auslender, A., ‘Stability in mathematical programming with nondifferentiable data’, SIAM J. Control Optim. (1984), 239–254.CrossRefGoogle Scholar
[4]Azé, D., ‘An inversion theorem for set-valued maps’, Bull. Austral. Math. Soc. 37 (1988), 411–414.Google Scholar
[5]Borwein, J.M., ‘Statility and regular point of inequality systems’, J. Optim. Theory Appl. 48 (1986), 9–52.Google Scholar
[6]Borwein, J.M. and Zhuang, D.M., ‘Verifiable necessary and sufficient conditions for openness and regularity for set-valued maps’, J. Math. Anal. Appl. 134 (1988), 441–459.CrossRefGoogle Scholar
[7]Chou, C.C., ‘Inverse function theorems for multifunctions in Prechet spaces’ (to appear).Google Scholar
[8]Clark, F., Optimization and nonsmooth analysis (Wiley Interscience, New York, 1983).Google Scholar
[9]De Blasi, F.S., ‘On the differentiability of multifunctions’, Pacific J. Math. 66 (1976), 67–81.Google Scholar
[10]Dolecki, S., ‘A general theory of necessary optimality conditions’, J. Math. Anal. Appl. 7 (1980), 267–308.CrossRefGoogle Scholar
[11]Ekeland, I., ‘Nonconvex minimization problems’, Bull. Amer. Math. Soc. 1 (1979), 443–474.CrossRefGoogle Scholar
[12]Frankowska, H., ‘An open mapping principle for set-valued maps’, J. Math. Anal. Appl. 127 (1987), 172–180.CrossRefGoogle Scholar
[13]Frankowska, H., ‘Some inverse mapping and open mapping theorems’ (to appear).Google Scholar
[14]Graves, L.M., ‘Some mapping theorems’, Duke Math. J. 17 (1950), 111–114.CrossRefGoogle Scholar
[15]Hiriart-Uruty, J.B., ‘Refinement of necessary optimality conditions in nondifferentiable programming’, Appl. Math. Optim. 5 (1979), 63–82.Google Scholar
[16]Ioffe, A.D., ‘Regular points of Lipschitz mapping’, Trans. Amer. Math. Soc. 251 (1979), 61–69.CrossRefGoogle Scholar
[17]Ioffe, A.D., ‘On the local surjection property’, Nonlinear Anal. 11 (1987), 565–592.CrossRefGoogle Scholar
[18]Jourani, A. and Thibault, L., ‘Approximate subdifferential and metric regularity: the finite dimensional case’, Math. Programming 47 (1990), 203–218.Google Scholar
[19]Jourani, A. and Thibault, L., ‘The use of metric graphical regularity in approximate sub-differential calculus rules in finite dimensions’, Optimization 21 (1990), 509–519.CrossRefGoogle Scholar
[20]Kantorovich, L.V. and Akilov, G.P., Functional analysis in normed spaces (Macmillan, New York, 1964).Google Scholar
[21]Penot, J.P., ‘On regularity conditions in mathematical programming’, Math. Programming Stud. 19 (1982), 167–199.CrossRefGoogle Scholar
[22]Penot, J.P., ‘Metric regularity, openness and Lipschitz behavior of multifunctions’, Nonlinear Anal. 13 (1989), 629–643.Google Scholar
[23]Robinson, S.M., ‘Stability theorems of systems of inequality part II: Differentiable nonlinear systems’, SIAM J. Numer. Anal. 13 (1976), 497–513.Google Scholar
[24]Robinson, S.M., ‘Regularity and stability for convex multivalued functions’, Math. Oper. Res. 1 (1976), 130–143.Google Scholar
[25]Rockafellar, R.T., ‘Lipschitzian properties of multifunctions’, Nonlinear Anal. 9 (1985), 867–885.CrossRefGoogle Scholar
[26]Rockafellar, R.T., ‘Extensions of subgradient calculus with application to optimization’, Nonlinear Anal. 9 (1985), 665–698.Google Scholar
[27]Thibault, L., ‘On subdifferential of optimal value function’, SIAM J. Control Optim. (to appear).Google Scholar
[28]Zowe, J. and Kurcyusz, S., ‘Regularity and stability for mathematical programming problems in Banach spaces’, Appl. Math. Optim. 5 (1979), 49–62.Google Scholar