Skip to main content
SpringerLink
Log in

Generalized Conjugation and Related Topics

  • Conference paper
Generalized Convexity and Fractional Programming with Economic Applications

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 345))

Abstract

An axiomatic approach to generalized conjugation theory, following [27], based on dualities between complete lattices, is outlined. Moreover, Toland-Singer type duality results for arbitrary conjugation operators, as well as a related formula for the Φ-conjugate of the difference of functions, which generalizes that of B. N. Pschenichnyi for the convex case, are studied in detail, examining some consequences for Moreau-Yosida and Lipschitz approximates of functions. Similar developments are made in connection with a duality formula due to M. Volle involving level set conjugations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson J.R., Duffin W.N., Trapp G.E., Parallel subtraction of matrices. Proceedings of the National Academy of Sciences of the U.S.A. 69, 1972, pp. 2530–2531.

    Article  Google Scholar 

  2. Attouch H., Variational convergence for functions and operators. Pitman, London, 1984.

    Google Scholar 

  3. Fermât Days 85: Mathematics for Optimization, North-Holland, Amsterdam - New York-Oxford-Tokyo, 1986, pp. 1–42.

    Google Scholar 

  4. Attouch H., Wets R.J.B., Isometries for the Legendre-Fenchel transform. Transactions of the American Mathematical Society 296, 1986, pp. 33–60.

    Article  Google Scholar 

  5. Balder E.J., An extension of duality-stability relations to non convex problems. SIAM Journal on Control and Optimization 15, 1977, pp.329–343.

    Google Scholar 

  6. Ben-Tai A., Ben-Israel A., F-convex functions: properties and applications. in: S. Schaible, W.T. Ziemba (Eds), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981, pp. 301–334.

    Google Scholar 

  7. Birkhoff G., Lattice Theory. American Mathematical Society. New York, 1948.

    Google Scholar 

  8. Brézis H., Opérateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert. North-Holland, Amsterdam-New York-Oxford-Tokyo, 1973.

    Google Scholar 

  9. Dolecki S., Kurcyusz S., Convexité généralisée et optimization. Comptes Rendus de Séances de l’Acedemie des Sciences de Paris 283, 1976, pp. 91–94.

    Google Scholar 

  10. Dolecki S., Kurcyusz S., On O-convexitv in extremal problems. SIAM Journal on Control and Optimization 16, 1978, pp. 277–300.

    Article  Google Scholar 

  11. Ellaia R., Hiriart-Urruty J.B., The conjugate of the difference of convex functions. Journal of Optimization Theory and Applications 49, 1986, pp. 493–498.

    Article  Google Scholar 

  12. Evers J.J., Van Maaren H., Duality principles in mathematics and their relation to conjugate functions. Nieuw Archiv voor Wiskunde 3, 1985, pp. 23–68.

    Google Scholar 

  13. Franchetti C., Singer I., Deviation and farthest points in normed linear spaces. Revue roumaine de mathématiques pures et appliquées 24, 1979, pp. 373–381.

    Google Scholar 

  14. Fujishige S., Theory of submodular programs: a Fenchel-tvpe minmax theorem and subgradients of sub modular functions. Mathematical Programming 29, 1984, pp. 142–155.

    Google Scholar 

  15. Gabay D., Minimizing the difference of two convex functions: Part I: Algorithms based on exact regularization. INRI A, working paper, 1982.

    Google Scholar 

  16. Hiriartürruty J.B., Extension of Lioschitz functions. Journal of Mathematical Analysis and Applications 77, 1980, pp. 539–554.

    Google Scholar 

  17. Hiriart-Urruty J.B., Lipschitz continuity of the approximate sub differential of a convex function. Matheplatica Scandinavica 47, 1980, pp. 123–134.

    Google Scholar 

  18. Hiriart-Urruty J.B., Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. in: J. Ponstein (Ed.), Convexity and Duality in Optimization, Springer Verlag, Berlin-Heidelberg-New York-Tokyo, 1985, pp. 37–70.

    Google Scholar 

  19. Hiriart-Ürruty J.B., A general formula on the conjugate of the difference of functions. Canadian Mathematical Bulletin 29, 1986, pp.482–485.

    Google Scholar 

  20. I20J Hiriart-Urruty J.B..Mazure M.L., Formulations variationeltes de l’addition parallèle et la soustraction parallèle d’opérateurs semi-définis positifs. Comptes Rendus de Séances de l’Academie des Sciences de Paris 302, 1986, pp.527–530.

    Google Scholar 

  21. Hiriart-Urruty J.B., Plazanet Ph., Moreau’s decomposition theorem revisited. Preprint, 1987.

    Google Scholar 

  22. J Hôrmander L., Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Arkivfur Mathematik 3, 1954, pp. 181–186.

    Google Scholar 

  23. Laurent P.J., Approximation et optimisation. Hermann, Paris, 1972.

    Google Scholar 

  24. Martinez-Legaz J.E., Conjugación asociada a un graf, in: Actas IX Jornadas Matemáticas Hispano-Lusas, Universidad de Salamanca, Salamanca, 1982, pp. 837–839.

    Google Scholar 

  25. Martinez-Legaz J.E., On lower sub differentiate functions, in: K.-H. Hoffmann, J.B. Hiriart-Urruty, C. Lemarechal, J. Zowe (Eds.), Trends in Mathematical Optimization. 4th French-German Conference on Optimization, Birkhâuser Verlag, Basel-Boston, 1988, pp. 197–232.

    Google Scholar 

  26. Martinez-Legaz J.E., Ouasiconvex duality theory by generalized conjugation methods. Optimization, to appear.

    Google Scholar 

  27. Martinez-Legaz J.E., Singer I., Dualities between complete lattices. INCREST, Preprint Series in Mathematics, n° 19/1987, (submitted).

    Google Scholar 

  28. Mazure M.L., La soustraction parallèle d’opérateurs interprétée comme déconvolution de formes quadratiques convexes. Optimization 18, 1987, pp. 465–484.

    Google Scholar 

  29. Me Shane E.J., Extension of range of functions. Bulletin of the American Mathematical Society 40, 1934, pp.837–842.

    Google Scholar 

  30. Moreau J.J., Fonctionelles convexes. Séminaire sur les Equation aux Dérivées Partielles II, Collège de France, Paris, 1966–1967.

    Google Scholar 

  31. Moreau J.J., Inf-convolution. sous-additivité. convexité des fonctions numériques. Journal de Mathématiques Pures et Appliquées 49, 1970, pp. 109–154.

    Google Scholar 

  32. Penot J.P., Voile M., On strongly convex and paraconvex dualities, this volume.

    Google Scholar 

  33. Poljak B.T., Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Mathematics Doklady 166, 1966, pp. 72–75.

    Google Scholar 

  34. Pshenichnyi B.N., Leçons sur les Jeux Différentiels. Cahiers de l’IRIA, vol. 4, 1971.

    Google Scholar 

  35. Pshenichnyi B.N., Daniline Y., Méthodes Numériques dans les Problèmes d’Extremum, Mir, Moscou, 1977.

    Google Scholar 

  36. Rockafellar R.T., Convex analysis. Princeton University Press, Princeton 1970.

    Google Scholar 

  37. Rockafellar R.T., Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM Journal on Control and Optimization 12, 1974, pp. 268–285.

    Google Scholar 

  38. Singer I., A Fenchel-Rochafellar type duality theorem for maximization. Bulletin of the Australian Mathematical Society 29, 1979, pp. 193–198.

    Google Scholar 

  39. Singer I., Maximization of lower semi-continuous convex functions on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems. Resultate der Mathematik 3, 1980, pp. 235–248.

    Google Scholar 

  40. Singer I., Generalized convexity, functional hulls and applications to conjugate duality in optimization, in: G. Hammer, D. Pallaschke (Eds.), Selected Topics in Operations Research and Mathematical Economics, Springer Verlag, Berlin-Heidelberg-New York, 1984, pp. 49–79.

    Chapter  Google Scholar 

  41. Singer I., Conjugation operators, ibidem, pp. 80–97.

    Google Scholar 

  42. Singer I., Some relations between dualities, polarities, coupling functional and conjugations. Journal of Mathematical Analysis and Applications 116, 1986, pp. 1–33.

    Google Scholar 

  43. Singer I., A general theory of dual optimization problems. Journal of Mathematical Analysis and Applications 116, 1986, pp.77-131.

    Google Scholar 

  44. Singer I., Generalizations of convex supremization duality. in: B.L. Lin, S. Simons (Eds.). Nonlinear and Convex Analysis, Marcel Dekker, New York, 1987, pp. 253–270.

    Google Scholar 

  45. Singer I., Infimal generators and dualities between complete lattices. Annali di Matematica Pura ed Applicata 148, 1987, pp. 289–352.

    Google Scholar 

  46. Stoer J., Witzgall C., Convexity and optimization in finite dimensions. I. Springer Verlag, Berlin-Heidelberg-New York, 1970.

    Google Scholar 

  47. Toland J.F., Duality in nonconvex optimization. Journal of Mathematical Analysis and Applications 66, 1978, pp. 399-415.

    Google Scholar 

  48. Toland J.F., A duality principie for nonconvex optimization and the calculus of variations. Archive for Rational Mechanics and Analysis 71, 1979, pp. 41–61.

    Google Scholar 

  49. Toland J.F., On sub differential calculus and duality in nonconvex optimization. Bulletin de la Société Mathématique de France 60, 1979, pp. 177–183

    Google Scholar 

  50. Vial J.P., Strong and weak convexity of sets and functions. Mathematics of Operations Research 8, 1983, pp. 279–311.

    Google Scholar 

  51. Voile M., Conjugaison par tranches. Annali di Matematica Pura ed Applicata 139, 1985, pp. 279–311.

    Google Scholar 

  52. Voile M., Contributions a la dualité en optimisation et a l’epiconvergence. Université de Pau et des Pays de l’Adour, Thèse, 1986.

    Google Scholar 

  53. Voile M., Concave duality: Application to problems dealing with difference of functions. Mathematical Programming 41, 1988, pp. 261–278.

    Google Scholar 

  54. Whitney HL, Analytic extensions of differentiable functions defined in closed sets. Transactions of the American Mathematical Society 36, 1934, pp. 63–89.

    Google Scholar 

Download references

Authors
  1. J. E. Martinez-Legaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Statistics and Applied Mathematics, University of Pisa, Via Ridolfi 10, I-56100, Pisa, Italy

    Alberto Cambini  & Laura Martein  & 

  2. Institute of Quantitative Methods, Bocconi University, Via Sarfatti 25, I-20136, Milano, Italy

    Erio Castagnoli

  3. Institute of Mathematics, University of Verona, Via dell’Artigliere 10, I-37129, Verona, Italy

    Piera Mazzoleni

  4. Graduate School of Management, University of California, 92521, Riverside, California, USA

    Siegfried Schaible

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Martinez-Legaz, J.E. (1990). Generalized Conjugation and Related Topics. In: Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P., Schaible, S. (eds) Generalized Convexity and Fractional Programming with Economic Applications. Lecture Notes in Economics and Mathematical Systems, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46709-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-46709-7_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52673-5

  • Online ISBN: 978-3-642-46709-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation