Second order differentiability of convex functions in Banach spaces

Borwein, Jonathan M.; Noll, Dominikus

doi:10.1090/S0002-9947-1994-1145959-4

Second order differentiability of convex functions in Banach spaces
HTML articles powered by AMS MathViewer

by Jonathan M. Borwein and Dominikus Noll

Trans. Amer. Math. Soc. 342 (1994), 43-81

DOI: https://doi.org/10.1090/S0002-9947-1994-1145959-4

PDF | Request permission

Abstract:

We present a second order differentiability theory for convex functions on Banach spaces.

References

Ralph Abraham, Jerrold E. Marsden, and Tudor S. Raţiu, Manifolds, tensor analysis, and applications, Global Analysis Pure and Applied: Series B, vol. 2, Addison-Wesley Publishing Co., Reading, Mass., 1983. MR 697563
A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 6 (1939), 3–35 (Russian). MR 0003051
N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147–190. MR 425608, DOI 10.4064/sm-57-2-147-190
Hédy Attouch, Familles d’opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl. (4) 120 (1979), 35–111 (French, with English summary). MR 551062, DOI 10.1007/BF02411939
Hédy Attouch and Roger J.-B. Wets, Isometries for the Legendre-Fenchel transform, Trans. Amer. Math. Soc. 296 (1986), no. 1, 33–60. MR 837797, DOI 10.1090/S0002-9947-1986-0837797-X
Victor Bangert, Analytische Eigenschaften konvexer Funktionen auf Riemannschen Mannigfaltigkeiten, J. Reine Angew. Math. 307(308) (1979), 309–324 (German). MR 534228, DOI 10.1515/crll.1979.307-308.309
A. Ben-Tal and J. Zowe, Directional derivatives in nonsmooth optimization, J. Optim. Theory Appl. 47 (1985), no. 4, 483–490. MR 818873, DOI 10.1007/BF00942193
Jonathan Borwein, Simon Fitzpatrick, and Petar Kenderov, Minimal convex uscos and monotone operators on small sets, Canad. J. Math. 43 (1991), no. 3, 461–476. MR 1118004, DOI 10.4153/CJM-1991-028-5
J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), no. 2, 517–527. MR 902782, DOI 10.1090/S0002-9947-1987-0902782-7

Sard’s theorem on Banach manifolds. Topics in differential geometry

Convex surfaces

Herbert Busemann and Willy Feller, Krümmungseigenschaften Konvexer Flächen, Acta Math. 66 (1936), no. 1, 1–47 (German). MR 1555408, DOI 10.1007/BF02546515
Jens Peter Reus Christensen, Measure theoretic zero sets in infinite dimensional spaces and applications to differentiability of Lipschitz mappings, Publ. Dép. Math. (Lyon) 10 (1973), no. 2, 29–39. MR 361770
V. F. Dem′yanov, C. Lemaréchal, and J. Zowe, Approximation to a set-valued mapping. I. A proposal, Appl. Math. Optim. 14 (1986), no. 3, 203–214. MR 867718, DOI 10.1007/BF01442236
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
M. Fabián, Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc. (3) 51 (1985), no. 1, 113–126. MR 788852, DOI 10.1112/plms/s3-51.1.113
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Simon Fitzpatrick and R. R. Phelps, Differentiability of the metric projection in Hilbert space, Trans. Amer. Math. Soc. 270 (1982), no. 2, 483–501. MR 645326, DOI 10.1090/S0002-9947-1982-0645326-5

Bounded approximants to monotone operators in Banach spaces

J.-B. Hiriart-Urruty, Lipschitz $r$-continuity of the approximate subdifferential of a convex function, Math. Scand. 47 (1980), no. 1, 123–134. MR 600082, DOI 10.7146/math.scand.a-11878
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
J.-B. Hiriart-Urruty and A. Seeger, Calculus rules on a new set-valued second order derivative for convex functions, Nonlinear Anal. 13 (1989), no. 6, 721–738. MR 998516, DOI 10.1016/0362-546X(89)90090-4
J.-B. Hiriart-Urruty and A. Seeger, The second-order subdifferential and the Dupin indicatrices of a nondifferentiable convex function, Proc. London Math. Soc. (3) 58 (1989), no. 2, 351–365. MR 977481, DOI 10.1112/plms/s3-58.2.351

Complément de Schur et sous-différentiel du second ordre d’une fonction convexe

Nobuyuki Kato, On the second derivatives of convex functions on Hilbert spaces, Proc. Amer. Math. Soc. 106 (1989), no. 3, 697–705. MR 960646, DOI 10.1090/S0002-9939-1989-0960646-4
F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22 (1976), no. 2, 130–185 (French). MR 0423155, DOI 10.1016/0022-1236(76)90017-3
Jean-Jacques Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273–299 (French). MR 201952
Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585. MR 298508, DOI 10.1016/0001-8708(69)90009-7
A. S. Nemirovskiĭ, The polynomial approximation of functions on Hilbert space, Funkcional. Anal. i Priložen. 7 (1973), no. 4, 88–89 (Russian). MR 0632032
Dominikus Noll, Second order differentiability of integral functionals on Sobolev spaces and $L^2$-spaces, J. Reine Angew. Math. 436 (1993), 1–17. MR 1207277, DOI 10.1515/crll.1993.436.1
Dominikus Noll, Generic Gâteaux-differentiability of convex functions on small sets, J. Math. Anal. Appl. 147 (1990), no. 2, 531–544. MR 1050225, DOI 10.1016/0022-247X(90)90368-P
Dominikus Noll, Generic Fréchet-differentiability of convex functions on small sets, Arch. Math. (Basel) 54 (1990), no. 5, 487–492. MR 1049204, DOI 10.1007/BF01188676
Dominikus Noll, Generalized second fundamental form for Lipschitzian hypersurfaces by way of second epi derivatives, Canad. Math. Bull. 35 (1992), no. 4, 523–536. MR 1191513, DOI 10.4153/CMB-1992-069-5
R. R. Phelps, Gaussian null sets and differentiability of Lipschitz map on Banach spaces, Pacific J. Math. 77 (1978), no. 2, 523–531. MR 510938
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
D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), no. 2, 312–345. MR 1058975, DOI 10.1016/0022-1236(90)90147-D
R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 167–184 (English, with French summary). MR 797269
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

Conjugate duality and optimization

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, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211–226. MR 479398, DOI 10.1016/0022-247X(77)90060-9
Josef Stoer and Christoph Witzgall, Convexity and optimization in finite dimensions. I, Die Grundlehren der mathematischen Wissenschaften, Band 163, Springer-Verlag, New York-Berlin, 1970. MR 0286498

Smooth approximation in Banach spaces

R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32–45. MR 196599, DOI 10.1090/S0002-9947-1966-0196599-8