Generalized second-order derivatives of convex functions in reflexive Banach spaces

Author:
Chi Ngoc Do

Journal:
Trans. Amer. Math. Soc. **334** (1992), 281-301

MSC:
Primary 49J52; Secondary 46G05

DOI:
https://doi.org/10.1090/S0002-9947-1992-1088019-1

MathSciNet review:
1088019

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Generalized second-order derivatives introduced by Rockafellar in finite-dimensional spaces are extended to convex functions in reflexive Banach spaces. Parallel results are shown in the infinite-dimensional case. A result that plays an important role in applications is that the generalized second-order differentiability is preserved under the integral sign.

**[1]**H. Attouch,*Variational convergence for functions and operators*, Pitman, London, 1984 MR**773850 (86f:49002)****[2]**H. Attouch, D. Azé and R. J.-B. Wets,*Convergence of convex-concave saddle functions*:*continuity properties of the Legendre-Fenchel transform*.*Applications to convex programming and mechanics*, Ann. Inst. Henri Poincaré**5**(1988), 537-572. MR**978671 (90g:90118)****[3]**A. BenTal and J. Zowe,*Necessary and sufficient conditions for a class of nonsmooth minimization problems*, Math. Programming**24**(1982), 70-91. MR**667940 (83m:90075)****[4]**R. W. Chaney,*Second-order sufficient conditions for nondifferentiable programming problems*, SIAM J. Control Optim.**20**(1982), 20-33. MR**642177 (83j:90077)****[5]**F. Clarke,*Optimization and nonsmooth analysis*, Wiley , New York, 1983. MR**709590 (85m:49002)****[6]**Chi N. Do,*Second-order nonsmooth analysis and sensitivity in optimization problems involving convex integral functionals*, PhD. Thesis, Univ. of Washington, 1989.**[7]**-,*Sensitivity analysis in optimal control problems with convex costs*, preprint.**[8]**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. MR**0481060 (58:1207)****[9]**J.-B. Hiriart-Urruty,*Calculus rules on the approximate second-order directional derivative of a convex function*, SIAM J. Control Optim.**22**(1984), 381-404. MR**739833 (85i:49023)****[10]**Jean-Luc Joly and François de Thelin,*Convergence of convex integrals in**spaces*, J. Math. Anal. Appl.**54**(1976), 230-244. MR**0412928 (54:1049)****[11]**J. L. Ndoutoume,*Calcul différentiel généralisé du second ordre*, preprint.**[12]**R. Phelps,*Convex functions, monotone operators and differentiability*, Springer-Verlag, 1989. MR**984602 (90g:46063)****[13]**R. T. Rockafellar,*First and second-order epi-differentiability in nonlinear programming*, Trans. Amer. Math. Soc.**307**(1988), 75-108. MR**936806 (90a:90216)****[14]**-,*Generalized second derivatives of convex functions and saddle functions*, Trans. Amer. Math. Soc.**28**(1990), 810-822. MR**1031242 (91b:90190)****[15]**-,*Integral functionals, normal integrands and measurable selections*, Nonlinear Operators and the Calculus of Variations, Springer-Verlag, 1976, pp. 157-207. MR**0512209 (58:23598)****[16]**-,*Proto-differentiability of set-valued mappings and its applications in optimization*, Analyse Non Linéaire (H. Attouch et al., eds.), Gauthier-Villars, Paris, 1989, pp. 449-482. MR**1019126 (90k:90140)****[17]**-,*Second-order optimality condition in nonlinear programming obtained by way of epiderivatives*, Math. Oper. Res.**14**(1989), 462-484. MR**1008425 (91b:49022)****[18]**G. Salinetti and R. J.-B. Wets,*Convergence of sequences of closed sets*, Topology Proc.**4**(1979), 149-157. MR**583698 (81j:54015)****[19]**-,*On the convergence of sequences of convex sets in finite dimensions*, SIAM Rev.**21**(1979), 18-33. MR**516381 (80h:52007)****[20]**A. Seeger,*Second-order directional derivatives in parametric optimization problems*, preprint. MR**931491 (89d:90217)****[21]**J. Sokolowski,*Differential stability of solutions to constrained optimization problems*, Appl. math. Optim., Springer-Verlag, 1985, pp. 97-115. MR**794173 (87i:49050)****[22]**E. H. Zarantonello,*Projections on convex sets in Hilbert space and spectral theory*, Contributions to Nonlinear Functional Analysis, 1971, pp. 237-424. MR**0388177 (52:9014)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
49J52,
46G05

Retrieve articles in all journals with MSC: 49J52, 46G05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1088019-1

Keywords:
Generalized second-order derivatives,
epi-convergence,
Mosco convergence,
epi-derivatives,
proto-derivatives,
integral functionals,
normal integrands

Article copyright:
© Copyright 1992
American Mathematical Society