Abstract

A number of rules for the calculus of subdifferentials of generalized convex functions are displayed. The subdifferentials we use are among the most significant for this class of functions, in particular for quasiconvex functions: we treat the Greenberg-Pierskalla's subdiffential and its relatives and the Plastria's lower subdifferential. We also deal with a recently introduced subdifferential constructed with the help of a generalized derivative. We emphasize the case of the sublevel-convolution, an operation analogous to the infimal convolution, which has proved to be of importance in the field of quasiconvex functions. We provide examples delineating the limits of the rules we provide.

Author information

Contact details are reproduced from the original publication and may be historical.

Jean-Paul Penot

Dép. de Mathématiques, Faculté des Sciences, Université de Pau, 64000 Pau, France

Constantin Zălinescu

Faculté de Mathématiques, Université "Al. I. Cuza", 6600 Iaşi, Romania

Keywords

  • Harmonic sum
  • incident derivative
  • lower subdifferential
  • normal cone
  • quasiconvex function
  • subdifferential set
  • sublevel-convolution

Mathematics Subject Classification

26B25, 90C26; 2651, 26A27

References

  1. D. Azé, M. Volle: A stability result in quasi-convex programming, J. Optim. Th. Appl. 67(1) (1990), 175–184.https://doi.org/10.1007/BF00939742Google Scholar
  2. J. M. Gutiérrez: Infragradientes y direcciones de decrecimiento, Rev. Real Acad. C. Ex., Fis. y Nat. Madrid 78 (1984), 523–532.Google Scholar
  3. J.-P. Penot: On regularity conditions in mathematical programming, Math. Progr. Study 19 (1982), 167–199.https://doi.org/10.1007/BFb0120988Google Scholar
  4. J.-P. Penot: Central and peripheral results in the study of marginal and performance functions, in: Mathematical Programming with Data Perturbations, A.V. Fiacco, ed. Lecture Notes in Pure and Applied Maths 195, Dekker, New York (1998) 305–337.https://doi.org/10.1201/9781003072119-14Google Scholar
  5. J.-P. Penot: Are generalized derivatives useful for generalized convex functions?, in: Generalized Convexity and Generalized Monotonicity, J.-P. Crouzeix et al. eds., Kluwer, Dordrecht (1998) 3–59.https://doi.org/10.1007/978-1-4613-3341-8_1Google Scholar
  6. J.-P. Penot: What is quasiconvex analysis?, Optimization (2000).https://doi.org/10.1080/02331930008844469Google Scholar
  7. J.-P. Penot, M. Volle: Inversion of real-valued functions and applications, Math. Methods Oper. Res. 34 (1990) 117–141.https://doi.org/10.1007/BF01415975Google Scholar
  8. J.-P. Penot, C. Zălinescu: Harmonic sum and duality, J. Convex Analysis 7 (2000) 95–113.https://doi.org/10.68381/jca07005Google Scholar
  9. J.-P. Penot, C. Zălinescu: Regularization of quasiconvex functions, in preparation.Google Scholar
  10. F. Plastria: Lower subdifferentiable functions and their minimization by cutting plane, J. Optim. Th. Appl. 46(1) (1994) 37–54.https://doi.org/10.1007/BF00938758Google Scholar
  11. A. Seeger, M. Volle: On a convolution operation obtained by adding level sets: classical and new results, RAIRO, Recherche Opérationnelle 29 (1995) 131–154.https://doi.org/10.1051/ro/1995290201311Google Scholar
  12. S. Traore, M. Volle: Quasi-convex conjugation and Mosco convergence, Ricerche Mat. 44(2) (1995) 369–388.Google Scholar
  13. S. Traore, M. Volle: On the level sum of two convex functions on Banach spaces, J. Convex Anal. 3(1) (1996) 141–151.https://doi.org/10.68381/jca03010Google Scholar
  14. M. Volle: Duality for the level sum of quasiconvex functions and applications, ESAIM: Control, Optimisation and Calculus of Variations, URL: http://www.emath.fr/cocv/Google Scholar
  15. C. Zălinescu: Mathematical Programming in Infinite Dimensional Normed Linear Spaces (Roumanian) Editura Academiei, Bucharest (1998), English translation to appear.Google Scholar
  16. D. T. Luc, M. Volle: Level sets under infimal convolution and level addition, J. Optim. Th. Appl. 94(3) (1997) 695–714.https://doi.org/10.1023/A:1022657102069Google Scholar

Suggested citation

J.-P. Penot, C. Zălinescu. “Elements of Quasiconvex Subdifferential Calculus.” Journal of Convex Analysis 7 (2000), No. 2, 243–269. https://doi.org/10.68381/jca07013

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2000. Rights now held by Banach Press.