Some types of convex functions on networks
DOI:
https://doi.org/10.33993/jnaat381-901Keywords:
\(E-d\)-convex functions, roughly \(d\)-convex functions, roughly \(E-d\)-convex functionsAbstract
We present and study some kinds of convex functions defined on undirected networks. The relations between these concepts are also presented. We adopt the definition of network as metric space used by Dearing P. M. and Francis R. L. in 1974.Downloads
References
Dearing, P.M. and Francis, R.L., A minimax location problem on a network, Transportation Science, 8, pp. 333-343, 1974, https://doi.org/10.1287/trsc.8.4.333 DOI: https://doi.org/10.1287/trsc.8.4.333
Dearing, P.M., Francis, R.L. and Lowe, T.J., Convex location problems on tree networks, Oper. Res., 24, pp. 628-634, 1976, https://doi.org/10.1287/opre.24.4.628 DOI: https://doi.org/10.1287/opre.24.4.628
de Groot, J., Some special metrics in general topology, Colloq. Math., 6, pp. 283-286, 1958, https://doi.org/10.4064/cm-6-1-283-286 DOI: https://doi.org/10.4064/cm-6-1-283-286
Hartwig H., Local Boundedness and Continuity of Generalized Convex Functions, Optimizations, 26, pp. 1-13, 1992, https://doi.org/10.1080/02331939208843838 DOI: https://doi.org/10.1080/02331939208843838
Hu, T.C., Klee, V. and Larman, D., Optimization of Globally Convex Functions, SIAM Journal on Control and Optimization, 27, pp. 1026-1047, 1989, https://doi.org/10.1137/0327055 DOI: https://doi.org/10.1137/0327055
Iacob, M.E., Optimization of d-convex functions on networks, Studia Univ. Babeş-Bolyai Math., XL, no. 3, pp. 45-58, 1995.
Iacob, M.E., Convexity, approximation and optimization on networks, Doctoral Dissertation, Babeş-Bolyai Univ., Cluj-Napoca, 1997.
Iacob, M.E. and Soltan, V.P., On geodetic characteristic of networks, Stud. Cerc. Mat., 46, no. 5, pp. 521-527, 1994.
Labbé, M., Essay in network location theory, Cahiers de Centre d'Etudes et de Recherche Oper., 27, nos. 1-2, pp. 7-130, 1985.
Marian, D., An Axiomatic Approach to the Theory of E-Convex Functions, Proceedings of the "Tiberiu Popoviciu" Itinerant Seminar of Functional Equations, Approximation and Convexity, Cluj-Napoca, May 22-26, pp. 97-106, 2001.
Marian, D., Direcţii ale analizei matematice tratate pe grafe şi pe reţele, Teză de doctorat, Universitatea "Babeş-Bolyai", Cluj-Napoca, 2002.
Marian, D., Generalized Convex Functions and Mathematical Analysis on Networks, Research on Theory of Allure, Approximation, Convexity and Optimization, Cluj-Napoca, pp. 183-206, 1999.
Menger, K., Untersuchungen über allgemeine Metrik I, II, III, Math. Ann., 100, pp. 75-163, 1928, https://doi.org/10.1007/bf01448840 DOI: https://doi.org/10.1007/BF01448840
Phu, H.X., Six Kinds of Roughly Convex Functions, Reprint 95-15, IWR, University of Heidelberg, 1995.
Phu, H.X., Strictly Roughly Convexlike Functions, Reprint 96-02, IWR, University of Heidelberg, 1996.
Phu, H.X., γ-Subdifferential and γ-Convexity of Functions on the Real Line, Applied Mathematics & Optimization, 27, pp. 145-160, 1993, https://doi.org/10.1007/bf01195979 DOI: https://doi.org/10.1007/BF01195979
Phu, H.X., γ-Subdifferential and γ-Convexity of Functions on a Normed Space, Journal of Optimization Theory and Aplications, 85, pp. 649-676, 1995, https://doi.org/10.1007/bf02193061 DOI: https://doi.org/10.1007/BF02193061
Phu, H.X. and Hai, N.N., Some Analytical Properties of γ-convex Function on the Real Line, Journal of Optimization Theory and Aplications, 91, no. 3, pp. 671-694, 1996, https://doi.org/10.1007/bf02190127 DOI: https://doi.org/10.1007/BF02190127
Phu, H.X., Hai, N.N. and An, P.T., Piecewise Constant Roughly Convex Functions, Reprint M-09/1995, Fakultat für Mathematik, Naturwissenschaften und Informatik, Technische Universität Gottbus, 1995.
Popoviciu, E., Teoreme de medie din analiza matematică şi legătura lor cu teoria interpolării, Dacia Cluj, 1972 (in Romanian).
Popoviciu, E., Analiză matematică, Litografia Univ. "Babeş-Bolyai", Cluj-Napoca, 1974 (in Romanian).
Popoviciu, T., Les fonctions convexes, Herman, Paris, 1945.
Rockafellar, R.T., Convex Analysis, Princeton University Press, Princeton, New-Jersey, 1970.
Söllner, B., Eigenschaften γ-grobconvexer Mengen und Funktionen. Diplomarbeit, Universitat Leipzig, 1991.
Soltan, P.S., Helly's theorem for d-convex sets (in Russian), Dokl. Akad. Nauk SSSR, 205, no. 3, pp. 537-539, 1972.
Soltan, V.P. and Chepoi, V.D., Some classes of d-convex functions in a graph, Soviet Math. Dokl., 28, no. 3, pp. 793-796, 1983.
Soltan, V.P. and Soltan, P.S., d-convex functions (in Russian), Dokl. Akad. Nauk SSSR, 249, pp. 555-558 1979.
Soltan, P.S. and Prisakaru, K.F., The Steiner problem on graphs, Soviet Math. Dokl., 12, no. 3, pp. 734-738, 1971.
Soltan, V.P., d-convexity in graphs, Soviet Math. Dokl., 28, no. 2, pp. 419-421, 1983.
Soltan, V.P., Introduction to the axiomatic theory of convexity (in Russian), Shtiinta, Kishinev, 1984.
Soltan, V.P., Some properties of d-convex functions I, Amer. Math. Soc. Transl., 134, no. 2, pp. 39-44, 1987, https://doi.org/10.1090/trans2/134/04 DOI: https://doi.org/10.1090/trans2/134/04
Soltan, V.P., Some properties of d-convex functions II, Amer. Math. Soc. Transl., 134, no. 2, pp. 45-51, 1987, https://doi.org/10.1090/trans2/134/05 DOI: https://doi.org/10.1090/trans2/134/05
Soltan, V.P., Metric convexity in graphs, Studia Univ. "Babeş-Bolyai" Math., 36, no. 4, pp. 3-43, 1991.
Soltan, V.P., An Axiomatic Approach to the Theory of Convex Functions, Soviet Math. Dokl., 22, no. 2, pp. 467-470, 1980.
Youness, E.A., E-Convex Sets, E-Convex Functions, and E-Convex Programming, Journal of Optimization Theory and Aplications, 102, no. 2, pp. 439-450, 1999, https://doi.org/10.1023/a:1021792726715 DOI: https://doi.org/10.1023/A:1021792726715
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.