Generalized secondorder derivatives of convex functions in reflexive Banach spaces
Author:
Chi Ngoc Do
Journal:
Trans. Amer. Math. Soc. 334 (1992), 281301
MSC:
Primary 49J52; Secondary 46G05
MathSciNet review:
1088019
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Abstract: Generalized secondorder derivatives introduced by Rockafellar in finitedimensional spaces are extended to convex functions in reflexive Banach spaces. Parallel results are shown in the infinitedimensional case. A result that plays an important role in applications is that the generalized secondorder differentiability is preserved under the integral sign.
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 [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 convexconcave saddle functions: continuity properties of the LegendreFenchel transform. Applications to convex programming and mechanics, Ann. Inst. Henri Poincaré 5 (1988), 537572. 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), 7091. MR 667940 (83m:90075)
 [4]
 R. W. Chaney, Secondorder sufficient conditions for nondifferentiable programming problems, SIAM J. Control Optim. 20 (1982), 2033. MR 642177 (83j:90077)
 [5]
 F. Clarke, Optimization and nonsmooth analysis, Wiley , New York, 1983. MR 709590 (85m:49002)
 [6]
 Chi N. Do, Secondorder 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), 615631. MR 0481060 (58:1207)
 [9]
 J.B. HiriartUrruty, Calculus rules on the approximate secondorder directional derivative of a convex function, SIAM J. Control Optim. 22 (1984), 381404. MR 739833 (85i:49023)
 [10]
 JeanLuc Joly and François de Thelin, Convergence of convex integrals in spaces, J. Math. Anal. Appl. 54 (1976), 230244. MR 0412928 (54:1049)
 [11]
 J. L. Ndoutoume, Calcul différentiel généralisé du second ordre, preprint.
 [12]
 R. Phelps, Convex functions, monotone operators and differentiability, SpringerVerlag, 1989. MR 984602 (90g:46063)
 [13]
 R. T. Rockafellar, First and secondorder epidifferentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), 75108. MR 936806 (90a:90216)
 [14]
 , Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 28 (1990), 810822. MR 1031242 (91b:90190)
 [15]
 , Integral functionals, normal integrands and measurable selections, Nonlinear Operators and the Calculus of Variations, SpringerVerlag, 1976, pp. 157207. MR 0512209 (58:23598)
 [16]
 , Protodifferentiability of setvalued mappings and its applications in optimization, Analyse Non Linéaire (H. Attouch et al., eds.), GauthierVillars, Paris, 1989, pp. 449482. MR 1019126 (90k:90140)
 [17]
 , Secondorder optimality condition in nonlinear programming obtained by way of epiderivatives, Math. Oper. Res. 14 (1989), 462484. MR 1008425 (91b:49022)
 [18]
 G. Salinetti and R. J.B. Wets, Convergence of sequences of closed sets, Topology Proc. 4 (1979), 149157. MR 583698 (81j:54015)
 [19]
 , On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), 1833. MR 516381 (80h:52007)
 [20]
 A. Seeger, Secondorder 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., SpringerVerlag, 1985, pp. 97115. 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. 237424. MR 0388177 (52:9014)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199210880191
PII:
S 00029947(1992)10880191
Keywords:
Generalized secondorder derivatives,
epiconvergence,
Mosco convergence,
epiderivatives,
protoderivatives,
integral functionals,
normal integrands
Article copyright:
© Copyright 1992 American Mathematical Society
