Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Limiting subhessians, limiting subjets
and their calculus

Authors: Alexander D. Ioffe and Jean-Paul Penot
Journal: Trans. Amer. Math. Soc. 349 (1997), 789-807
MSC (1991): Primary 28A15, 46G05; Secondary 26A24, 26A27
MathSciNet review: 1373640
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study calculus rules for limiting subjets of order two. These subjets are obtained as limits of sequences of subjets, a subjet of a function $f$ at some point $x$ being the Taylor expansion of a twice differentiable function which minorizes $f$ and coincides with $f$ at $x$. These calculus rules are deduced from approximate (or fuzzy) calculus rules for subjets of order two. In turn, these rules are consequences of delicate results of Crandall-Ishii-Lions. We point out the similarities and the differences with the case of first order limiting subdifferentials.

References [Enhancements On Off] (What's this?)

  • 1. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Basel, 1990. MR 91d:49001
  • 2. R. Cominetti and R. Correa, Sur une dérivée du second ordre en analyse non différentiable, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 861-864. MR 87m:49040
  • 3. -, A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim. 28 (4) (1990), 789-809. MR 91h:49017
  • 4. M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Differential and Integral Equations 3 (1990), 1001-1014.
  • 5. M. G. Crandall, H. Ishii, and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1-67. MR 92j:35050
  • 6. J.-B. Hiriart-Urruty and Ph. Plazanet, Moreau's theorem revisited, Analyse Non Linéaire (Perpignan, 1987), Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), suppl., 325-338. MR 91b:58017
  • 7. J.-B. Hiriart-Urruty, J.-J. Strodiot and V. H. Nguyen, Generalized hessian matrix and second order conditions for problems with $C^{1,1}$ data, Appl. Math. Optim. 11 (1984), 43-56. MR 85g:49031
  • 8. A. D. Ioffe, Second order conditions in nonlinear nonsmooth problems of semi-infinite programming, Semi-infinite Programming and Applications, Lecture Notes in Economics and Math. Systems, vol. 215, Springer-Verlag, Berlin, 1983, pp. 262-280. MR 84m:49060
  • 9. -, Approximate subdifferentials and applications. 1: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389-416; 288 (1985), 429. MR 84m:49029; MR 86d:49023
  • 10. -, Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175-192. MR 91i:46045
  • 11. R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), 1-27. MR 89a:35038
  • 12. A. Ya. Kruger, Properties of generalized differentials, Sibirsk. Mat. Zhurnal 26 (1985), no. 6, 54-66; English transl., Siberian Math. J. 26 (1985), 822-832. MR 87d:90147
  • 13. L. A. Lusternik and V. J. Sobolev, Elements of functional analysis, Hindustan, Delhi, and Wiley, New York, 1974. MR 50:2852
  • 14. B. Sh. Mordukhovich, Maximum principle in the problem of time-optimal response with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960-969. MR 58:7284
  • 15. -, Nonsmooth analysis with nonconvex generalized differentials and adjoint mappings, Dokl. Akad. Nauk BSSR 28 (1984), 976-979. (Russian) MR 86c:49018
  • 16. J.-P. Penot, Regularity conditions in mathematical programming, Math. Programming Study 19 (1982), 167-199. MR 84d:90095
  • 17. -, Subhessians, superhessians and conjugation, Nonlinear Anal. 23 (1994), 689-702. MR 95h:49025
  • 18. J. P. Penot and M. Volle, On strongly convex and paraconvex dualities, Generalized Convexity and Fractional Programming with Economic Applications, A. Cambini et al. (eds.), Lecture Notes in Economics and Math. Systems, vol. 345, Springer-Verlag, Berlin, 1990, pp. 198-218. MR 92h:49041
  • 19. R. T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization, Progress in Nondifferentiable Optimization (E. Nurminski, ed.), IIASA Collaborative Proc. Ser. CP-82, 8, Internat. Inst. Appl. Systems Analysis, Laxenburg, Austria, 1982, pp. 125-144. MR 85e:90069
  • 20. -, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), 75-108. MR 90a:90216
  • 21. S. Rolewicz, On paraconvex multifunctions, Proc. III Sympos. Oper. Research, Mannheim, 1978, 539-546. MR 80i:49025

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 28A15, 46G05, 26A24, 26A27

Retrieve articles in all journals with MSC (1991): 28A15, 46G05, 26A24, 26A27

Additional Information

Alexander D. Ioffe
Affiliation: Department of Mathematics, Technion, 32000 Haifa, Israel

Jean-Paul Penot
Affiliation: Départment de Mathématiques, CNRS URA 1204, Faculté des Sciences, Av. de l’Université, 64000 Pau, France
Email: jean-paul.penot@univ.pau-fr

Received by editor(s): August 3, 1994
Received by editor(s) in revised form: September 5, 1995
Additional Notes: The first author’s research was supported in part by the U.S.-Israel Binational Science Foundation, under grant no. 90-00455
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society