Second-Order Epi-Derivatives of Composite Functionals
- A.B. Levy
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
We compute two-sided second-order epi-derivatives 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 real-valued 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 second-order upper epi-derivative that mirrors a formula for a second-order lower epi-derivative from , and the two-sided 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.
- J.F. Bonnans and R. Cominetti, Perturbed optimization in Banach spaces I: A general theory based on a weak directional constraint qualification, SIAM J. Control and Optimization 34 (1996) 1151–1171.
- J.R. Bonnans, R. Cominetti and A. Shapiro, Second order optimality conditions based on parabolic second order tangent sets, SIAM J. Optimization 9 (1999) 466–492.
- R. Cominetti, On pseudo–differentiability, Transactions of the American Mathematical Society 324 (1991) 843–865.
- A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29 (1977) 615–631.
- A.D. Ioffe, Variational analysis of a composite function: A formula for the lower second–order epiderivative, Journal of Mathematical Analysis and Applications 160 (1991) 379–405.
- A.B. Levy, Sensitivity of solutions to variational inequalities on Banach spaces, SIAM J. Control and Optimization 38 (1999) 50–60.
- A.B. Levy, R. Poliquin and L. Thibault, Partial extensions of Attouch's theorem with application to proto–derivatives of subgradient mappings, Transactions of the American Mathematical Society 347 (1995) 1269–1294.
- F. Mignot, Contrôle dans les inéquations variational elliptiques, Journal of Functional Analysis 22 (1976) 130–185.
- R.T. Rockafellar, First–and second–order epi–differentiability in nonlinear programming, Transactions of the American Mathematical Society 307 (1988) 75–107.
- R.T. Rockafellar, Second–order optimality conditions in nonlinear programming obtained by way of epi–derivatives, Mathematics of Operations Research 14 (1989) 462–484.
- R.T. Rockafellar and R.J.B. Wets, Variational Analysis (Springer, 1998).
- M. Torki, First–and second–order epi–differentiability in eigenvalue optimization, Journal of Mathematical Analysis and Applications 234 (1999) 391–416.
- Second-Order Epi-Derivatives of Composite Functionals
Annals of Operations Research
Volume 101, Issue 1-4 , pp 267-281
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- second-order epi-derivative
- twice Mosco epi-differentiability
- convex-C 2 composite function
- Industry Sectors
- A.B. Levy (1)
- Author Affiliations
- 1. Department of Mathematics, Bowdoin College, Brunswick, ME, 04011, USA