Abstract
This work is devoted to a systematic study of the inversion of nondecreasing one variable extended real-valued functions. Its results are preparatory for a new duality theory for quasiconvex problem [6]. However the question arises in a variety of situations and as such deserves a separate treatment. Applications to topology, probability theory, monotone rearrangements, convex analysis are either pointed out or sketched.
Zusammenfassung
In dieser Arbeit wird systematisch die Umkehrung monoton nichtfallender Funktionenf: ℝ → ℝ ∪ {−∞, +∞} studiert. Die Ergebnisse bilden die Grundlage für eine neue Dualitätstheorie quasikonvexer Probleme [6]. Da jedoch die Fragestellung bei einer ganzen Anzahl weiterer Situationen auftritt, verdient sie eine gesonderte Behandlung. Anwendungen in der Topologie, Wahrscheinlichkeitstheorie, monotonen Umordnungen und in der konvexen Analysis werden aufgezeigt und skizziert.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Crouzeix JP (1977) Contribution à l'étude des fonctions quasi-convexes. Thèse Université de Clermont-Ferrand II
Crouzeix JF (1981) A duality framework in quasiconvex programming. In: Generalized concavity in optimization and economics. Academic Press, New York, pp 207–225
Dacunha-Castelle D, Duflo M (1982) Probabilités et statistiques I. Exercices. Masson, Paris
Martinez Legaz JE (1983) A generalized concept of conjugation. In: Hiriart-Urruty JB et al. (ed) Optimization theory and algorithms. Dekker, New York, pp 45–59
Mossino J (1984) Inégalités isopérimétriques et applications en physique. Hermann, Paris
Penot J-P, Volle M. On quasiconvex duality. To appear in Math Oper Research
Rockafellar RT (1967) Convex programming and systems of elementary monotone relations. J Math Anal and Appl 19:543–564
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton, New Jersey
Rockafellar RT (1984) Monotropic optimization. Wiley, New York
Sherwood H, Taylor MD (1974) Some P.M. Structure on the set of distribution functions. Rev Roum Math pures et appl XIX/10:1251–1260
Talenti G (1976) Elliptic equations and rearrangements. Ann delle Scuola Normale Superiore di Pisa serie 4, n∘ 3:697–718
Young WH (1912) On classes of summable functions and their Fourier series. Proc Ry Soc (A) 87:225–229
Zaanen AC (1983) Riesz spaces II. North-Holland Publishing Company, Amsterdam
Zalinescu C (1983) On uniformly convex functions. J Math Anal and Appl 95:344–374
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Penot, J.P., Volle, M. Inversion of real-valued functions and applications. ZOR - Methods and Models of Operations Research 34, 117–141 (1990). https://doi.org/10.1007/BF01415975
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01415975