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SecondOrder EpiDerivatives of Composite Functionals
 A.B. Levy
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We compute twosided secondorder epiderivatives for certain composite functionals f=g○F where F is a C ^{1} mapping between two Banach spaces X and Y, and g is a convex extended realvalued function on Y. These functionals include most essential objectives associated with smooth constrained minimization problems on Banach spaces. Our proof relies on our development of a formula for the secondorder upper epiderivative that mirrors a formula for a secondorder lower epiderivative from [7], and the twosided results we obtain promise to support a more precise sensitivity analysis of parameterized optimization problems than has been previously possible.
 J.–P. Aubin and I. Ekeland, Applied Nonlinear Analysis (Wiley, 1984).
 L. Barbet, Stability and differential sensitivity of optimal solutions of parameterized variational inequalities in infinite dimensions, in: Parametric Optimization and Related Topics 9 (Akademie–Verlag, 1995) pp. 25–41.
 Bonnans, J.F., Cominetti, R. (1996) Perturbed optimization in Banach spaces I: A general theory based on a weak directional constraint qualification. SIAM J. Control and Optimization 34: pp. 11511171
 Bonnans, J.R., Cominetti, R., Shapiro, A. (1999) Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optimization 9: pp. 466492
 Cominetti, R. (1991) On pseudo–differentiability. Transactions of the American Mathematical Society 324: pp. 843865
 Haraux, A. (1977) How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29: pp. 615631
 Ioffe, A.D. (1991) Variational analysis of a composite function: A formula for the lower second–order epiderivative. Journal of Mathematical Analysis and Applications 160: pp. 379405
 Levy, A.B. (1999) Sensitivity of solutions to variational inequalities on Banach spaces. SIAM J. Control and Optimization 38: pp. 5060
 Levy, A.B., Poliquin, R., Thibault, L. (1995) Partial extensions of Attouch's theorem with application to proto–derivatives of subgradient mappings. Transactions of the American Mathematical Society 347: pp. 12691294
 Mignot, F. (1976) Contrôle dans les inéquations variational elliptiques. Journal of Functional Analysis 22: pp. 130185
 Rockafellar, R.T. (1988) First–and second–order epi–differentiability in nonlinear programming. Transactions of the American Mathematical Society 307: pp. 75107
 Rockafellar, R.T. (1989) Second–order optimality conditions in nonlinear programming obtained by way of epi–derivatives. Mathematics of Operations Research 14: pp. 462484
 R.T. Rockafellar and R.J.B. Wets, Variational Analysis (Springer, 1998).
 Torki, M. (1999) First–and second–order epi–differentiability in eigenvalue optimization. Journal of Mathematical Analysis and Applications 234: pp. 391416
 Title
 SecondOrder EpiDerivatives of Composite Functionals
 Journal

Annals of Operations Research
Volume 101, Issue 14 , pp 267281
 Cover Date
 20010101
 DOI
 10.1023/A:1010993128564
 Print ISSN
 02545330
 Online ISSN
 15729338
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 secondorder epiderivative
 twice Mosco epidifferentiability
 convexC 2 composite function
 Industry Sectors
 Authors

 A.B. Levy ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Bowdoin College, Brunswick, ME, 04011, USA