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

Do, Chi Ngoc

doi:10.1090/S0002-9947-1992-1088019-1

Generalized second-order derivatives of convex functions in reflexive Banach spaces
HTML articles powered by AMS MathViewer

by Chi Ngoc Do PDF

Trans. Amer. Math. Soc. 334 (1992), 281-301 Request permission

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.

References

H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
Dominique Azé, Hédy Attouch, and Roger J.-B. Wets, Convergence of convex-concave saddle functions: applications to convex programming and mechanics, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 6, 537–572 (English, with French summary). MR 978671
A. Ben-Tal and J. Zowe, Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems, Math. Programming 24 (1982), no. 1, 70–91. MR 667940, DOI 10.1007/BF01585095
R. W. Chaney, Second-order sufficiency conditions for nondifferentiable programming problems, SIAM J. Control Optim. 20 (1982), no. 1, 20–33. MR 642177, DOI 10.1137/0320004
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590

Second-order nonsmooth analysis and sensitivity in optimization problems involving convex integral functionals

Sensitivity analysis in optimal control problems with convex costs

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), no. 4, 615–631. MR 481060, DOI 10.2969/jmsj/02940615
J.-B. Hiriart-Urruty, Calculus rules on the approximate second-order directional derivative of a convex function, SIAM J. Control Optim. 22 (1984), no. 3, 381–404. MR 739833, DOI 10.1137/0322025
Jean-Luc Joly and François de Thelin, Convergence of convex integrals in ${\cal L}^{p}$ spaces, J. Math. Anal. Appl. 54 (1976), no. 1, 230–244. MR 412928, DOI 10.1016/0022-247X(76)90247-X

Calcul différentiel généralisé du second ordre

Robert R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR 984602, DOI 10.1007/BFb0089089
R. T. Rockafellar, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), no. 1, 75–108. MR 936806, DOI 10.1090/S0002-9947-1988-0936806-9
R. T. Rockafellar, Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 322 (1990), no. 1, 51–77. MR 1031242, DOI 10.1090/S0002-9947-1990-1031242-0
R. Tyrrell Rockafellar, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Lecture Notes in Math., Vol. 543, Springer, Berlin, 1976, pp. 157–207. MR 0512209
R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. suppl., 449–482. Analyse non linéaire (Perpignan, 1987). MR 1019126, DOI 10.1016/S0294-1449(17)30034-3
R. Tyrrell Rockafellar, Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. Oper. Res. 14 (1989), no. 3, 462–484. MR 1008425, DOI 10.1287/moor.14.3.462
Gabriella Salinetti and Roger J.-B. Wets, Convergence of sequences of closed sets, Topology Proc. 4 (1979), no. 1, 149–158 (1980). MR 583698
Gabriella Salinetti and Roger J.-B. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), no. 1, 18–33. MR 516381, DOI 10.1137/1021002
Alberto Seeger, Second order directional derivatives in parametric optimization problems, Math. Oper. Res. 13 (1988), no. 1, 124–139. MR 931491, DOI 10.1287/moor.13.1.124
Jan Sokołowski, Differential stability of solutions to constrained optimization problems, Appl. Math. Optim. 13 (1985), no. 2, 97–115. MR 794173, DOI 10.1007/BF01442201
Eduardo H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 237–341. MR 0388177