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)

 
 

 

Prox-regular functions in
variational analysis


Authors: R. A. Poliquin and R. T. Rockafellar
Journal: Trans. Amer. Math. Soc. 348 (1996), 1805-1838
MSC (1991): Primary 49A52, 58C06, 58C20; Secondary 90C30
DOI: https://doi.org/10.1090/S0002-9947-96-01544-9
MathSciNet review: 1333397
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-$ \mathcal{C}^{2}$ functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.


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

  • 1. H. Attouch, Variational convergence for functions and operators, Pitman, 1984. MR 86f:49002
  • 2. H. Attouch, and R. J-B Wets, Epigraphical analysis, Analyse non linéaire 6 (1989), 73--100. MR 91g:90161
  • 3. J.V. Burke and R. A. Poliquin, Optimality conditions for non-finite convex composite functions, Mathematical Programming 57 (1992), 103--120. MR 93g:90066
  • 4. F. H. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Math., vol. 5, SIAM Publications, Philadelphia, 1990 (originally published in 1983). MR 85m:49002
  • 5. ------, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics 57, SIAM Publications, Philadelphia, 1989. MR 91j:49001
  • 6. F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-${\mathcal C} ^{2}$ property, preprint (1994).
  • 7. R. Cominetti, On Pseudo-differentiability, Trans.Amer. Math. Soc. 324 (1991), 843--865. MR 91h:26009
  • 8. C. Do, Generalized second derivatives of convex functions in reflexive Banach spaces, Trans. Amer. Math. Soc. 334 (1992), 281--301. MR 93a:49011
  • 9. A. D. Ioffe, Variational analysis of a composite function: a formula for the lower second order epi-derivative, J. Math. Anal. Appl.160 (1991), 379--405. MR 92m:46061
  • 10. A. Levy, R. A. Poliquin and L. Thibault, Partial extension of Attouch's theorem with applications to proto-derivatives of subgradient mappings, Trans. Amer. Math. Soc. 347 (1995), 1269--1294. MR 95:07
  • 11. A. Levy and R. T. Rockafellar, Variational conditions and the proto-differentiation of partial subgradient mappings, Nonlinear Anal. Th. Meth. Appl. (to appear).
  • 12. R. A. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Analysis, Theory, Methods & Applications, 14 (1990), 305-317. MR 91b:90155
  • 13. ------, Proto-differentiation of subgradient set-valued mappings, Canadian J. Math. 42 (1990), 520--532. MR 91g:49007
  • 14. ------, Integration of subdifferentials of nonconvex functions, Nonlinear Anal. Th. Meth. Appl. 17 (1991), 385--398. MR 92j:49008
  • 15. ------, An extension of Attouch's Theorem and its application to second-order epi-differentiation of convexly composite functions, Trans. Amer. Math. Soc. 332 (1992), 861--874. MR 93a:49013
  • 16. R. A. Poliquin and R. T. Rockafellar, Amenable functions in optimization, Nonsmooth Optimization Methods and Applications (F. Giannessi, ed.), Gordon and Breach, Philadelphia, 1992, pp. 338--353. MR 95d:49033
  • 17. ------, A calculus of epi-derivatives applicable to optimization, Canadian J. Math. 45 (1993), 879--896. MR 94d:49023
  • 18. ------, Proto-derivative formulas for basic subgradient mappings in mathematical programming, Set-Valued Analysis, 2 (1994), 275--290. MR 95c:49024
  • 19. ------, Generalized Hessian properties of regularized nonsmooth functions, SIAM J. Optimization (to appear).
  • 20. ------, Second-order nonsmooth analysis in nonlinear programming, Recent Advances in Nonsmooth Optimization (D. Du, L. Qi and R. Womersley, eds.), World Scientific Publishers, 1995, pp. 322--350.
  • 21. R. A. Poliquin, J. Vanderwerff and V. Zizler, Convex composite representation of lower semicontinuous functions and renormings, C.R. Acad.Sci. Paris 317 Série I (1993), 545--549. MR 94i:46102
  • 22. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. MR 43:445
  • 23. ------, Local boundedness of nonlinear monotone operators, Michigan Math. 16 (1969), 397--407. MR 40:6229
  • 24. ------, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. of Op. Res. 6 (1981), 424--436. MR 83m:90088
  • 25. ------, Favorable classes of Lipschitz continuous functions in subgradient optimization, Progress in Nondifferentiable Optimization (E. Nurminski, ed.), IIASA Collaborative Proceedings Series, International Institute of Applied Systems Analysis, Laxenburg, Austria, 1982, pp. 125-144. MR 85c:90069
  • 26. ------, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré: Analyse non linéaire 2 (1985), 167--184. MR 87c:49021
  • 27. ------, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988, 75--107). MR 90a:90216
  • 28. ------, 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 90k:90140
  • 29. ------, Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. of Oper.Research 14 (1989), 462--484. MR 91b:49022
  • 30. ------, Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 320 (1990), 810--822. MR 91b:90190
  • 31. ------, Nonsmooth analysis and parametric optimization, Methods of Nonconvex Analysis (A. Cellina, ed.), vol. 1446, Springer-Verlag Lecture Notes in Math, 1990, pp. 137--151. MR 91i:49016
  • 32. ------, Lagrange multipliers and optimality, SIAM Review 35 (1993), 183--238. MR 94h:49004
  • 33. G. Salinetti and R. J-B Wets, On the relation between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), 211--226. MR 57:18828
  • 34. L. Thibault and D. Zagrodny, Integration of subdifferentials of lower semicontinuous functions, J. Math. Anal. Appl. 189 (1995), 33--58.
  • 35. J. P. Vial, Strong and weak convexity of sets and functions, Math of Oper. Res. 8 (1983), 231--259. MR 84m:90107

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 49A52, 58C06, 58C20, 90C30

Retrieve articles in all journals with MSC (1991): 49A52, 58C06, 58C20, 90C30


Additional Information

R. A. Poliquin
Affiliation: Deptartment of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: rene@fenchel.math.ualberta.ca

R. T. Rockafellar
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: rtr@math.washington.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01544-9
Keywords: Prox-regularity, amenable functions, primal-lower-nice functions, proximal mappings, Moreau envelopes, regularization, subgradient mappings, nonsmooth analysis, variational analysis, proto-derivatives, second-order epi-derivatives, Attouch's theorem
Received by editor(s): December 21, 1994
Received by editor(s) in revised form: June 7, 1995
Additional Notes: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author and by the National Science Foundation under grant DMS–9200303 for the second author.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society