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A chain rule for nonsmooth composite functions via minimisation

Published online by Cambridge University Press:  17 April 2009

D. Ralph
D. Ralph
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Vic. 3052, Australia
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Abstract

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Nonsmooth calculus using the approximate subdifferential of Mordukhovich and loffe admits a sharper chain rule, hence sharper applications in optimisation, than does the generalised gradient of Clarke. We observe, however, that at a local minimum point of the composition of nonsmooth vector valued and real valued functions, the generalised gradient admits a special, relatively sharp chain rule, that yields sharper results than have been seen before in the context of the generalised gradient.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 1 , February 1994 , pp. 129 - 137
Copyright
Copyright © Australian Mathematical Society 1994

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.Google Scholar
[3]Clarke, F.H., Optimization and nonamooth analysis (Wiley Interscience, New York, 1983).Google Scholar
[4]Fletcher, R., Practical methods of optimization, 2nd ed. (John Wiley and Sons, New York, 1987).Google Scholar
[5]Glover, B.M., ‘Locally compactly Lipschitzian mappings in infinite dimensional programming’, Bull. Austral. Mathem. Soc. 47 (1993), 395–406.Google Scholar
[6]Glover, B.M. and Ralph, D., ‘First order approximations to nonsmooth mappings with applications to metric regularity’, in Preprint Series No. 3 — 1993 (Department of Mathematics, University of Melbourne, 1993).Google Scholar
[7]Ioffe, A.D., ‘Regular points of Lipschitz mappings’, Trans. Amer. Math. Soc. 251 (1979), 61–69.Google Scholar
[8]Ioffe, A.D., ‘Nonsmooth analysis: differential calculus of nondifferentiable mappings’, Trans. Amer. Math. Soc. 266 (1981), 1–56.Google Scholar
[9]Ioffe, A.D., ‘Approximate subdifferentials and applications II: Functions on locally convex spaces’, Mathematika 33 (1986), 111–128.CrossRefGoogle Scholar
[10]Ioffe, A.D., ‘Approximate subdifferentials and applications III: The metric theory’, Mathematika 36 (1989), 1–38.Google Scholar
[11]Jourani, A. and Thibault, L., ‘Approximations and metric regularity in mathematical programming in Banach spaces’, Math. Oper. Res. (1992) (to appear).Google Scholar
[12]Jourani, A. and Thibault, L., ‘Approximate subdifferential of composite functions’, Bull. Austral. Math. Soc. 47 (1992), 443–455.Google Scholar
[13]Mordukhovich, B.S., ‘Nonsmooth analysis with nonconvex generalized differentials and dual maps’, Dokl. Akad. Nauk. BSSR 28 (1984), 976–979.Google Scholar
[14]Ralph, D., Rank-1 Support functionals and the generalized Jacobian, piecewise linear homeomorphisms, Ph.D Thesis (Computer Sciences Department, University of Wisconsin, Madison, 1990).Google Scholar
[15]Robertson, A.P. and Robertson, W., Topological vector spaces (Cambridge University Press, 1973).Google Scholar
[16]Robinson, S.M., ‘Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems’, SIAM J. Numer. Anal. 13 (1976), 497–513.CrossRefGoogle Scholar
[17]Thibault, L., ‘On generalized differentials and subdifferentials of Lipschitz vector-valued functions’, Nonlinear Analisis Theory, Methods and Applications 6 (1982), 1037–1053.Google Scholar