Advertisement

Metrical Pareto Efficiency and Monotone EVP

  • Mihai TuriniciEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 113)

Abstract

A uniform version is established for normed Pareto efficient point results in Isac [Comb. Global Optim., pp. 133–144, World Sci. Publ., 2002]. Its basic tool is the monotone variant of Ekeland’s variational principle obtained in Turinici [An. Şt. UAIC Iaşi, 36 (1990), 329–352].

Keywords

Inf-proper lsc function Dependent Choice Monotone variational principle Generalized metric/uniform space Pareto efficiency Super-additive function 

References

  1. 1.
    M. Altman, A generalization of the Brezis-Browder principle on ordered sets. Nonlinear Anal. 6, 157–165 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J.S. Bae, E.W. Cho, S.H. Yeom, A generalization of the Caristi-Kirk fixed point theorem and its applications to mapping theorems. J. Korean Math. Soc. 31, 29–48 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    P. Bernays, A system of axiomatic set theory: Part III. Infinity and enumerability analysis. J. Symb. Log. 7, 65–89 (1942)CrossRefzbMATHGoogle Scholar
  4. 4.
    C.E. Blair, The Baire category theorem implies the principle of dependent choice. Bull. Acad. Pol. Sci. (Sér. Math. Astronom. Phys.) 10, 933–934 (1977)MathSciNetzbMATHGoogle Scholar
  5. 5.
    N. Bourbaki, General Topology (Chapters 1 –4) (Springer, Berlin, 1989)CrossRefzbMATHGoogle Scholar
  6. 6.
    H. Brezis, F.E. Browder, A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 21, 355–364 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Brøndsted, On a lemma of Bishop and Phelps. Pac. J. Math. 55, 335–341 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Brøndsted, Fixed points and partial orders. Proc. Am. Math. Soc. 60, 365–366 (1976)MathSciNetzbMATHGoogle Scholar
  9. 9.
    N. Brunner, Topologische Maximalprinzipien. Z. Math. Logik Grundl. Math. 33, 135–139 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    O. Cârjă, M. Necula, I.I. Vrabie, Viability, Invariance and Applications. North Holland Mathematics Studies, vol. 207 (Elsevier B. V., Amsterdam, 2007)Google Scholar
  11. 11.
    P.J. Cohen, Set Theory and the Continuum Hypothesis (Benjamin, New York, 1966)zbMATHGoogle Scholar
  12. 12.
    V. Conserva, S. Rizzo, Maximal elements in a class of order complete metric subspaces. Math. Jpn. 37, 515–518 (1992)MathSciNetzbMATHGoogle Scholar
  13. 13.
    R. Cristescu, Topological Vector Spaces (Noordhoff International Publishers, Leyden, 1977)zbMATHGoogle Scholar
  14. 14.
    S. Dancs, M. Hegedus, P. Medvegyev, A general ordering and fixed-point principle in complete metric space. Acta Sci. Math. (Szeged) 46, 381–388 (1983)MathSciNetzbMATHGoogle Scholar
  15. 15.
    J. Dodu, M. Morillon, The Hahn-Banach property and the Axiom of Choice. Math. Log. Q. 45, 299–314 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    I. Ekeland, Nonconvex minimization problems. Bull. Am. Math. Soc. (New Series) 1, 443–474 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J.X. Fang, The variational principle and fixed point theorems in certain topological spaces. J. Math. Anal. Appl. 202, 398–412 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    C. Gerth (Tammer), P. Weidner, Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)Google Scholar
  19. 19.
    A. Goepfert, C. Tammer, A new maximal point theorem. Z. Anal. Anwend. (J. Analysis Appl.) 14, 379–390 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. Goepfert, C. Tammer, C. Zălinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Anal. 39, 909–922 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Goepfert, H. Riahi, C. Tammer, C. Zălinescu, Variational Methods in Partially Ordered Spaces. Canadian Mathematical Society Books in Mathematics, vol. 17 (Springer, New York, 2003)Google Scholar
  22. 22.
    R. Goldblatt, On the role of the Baire Category theorem and Dependent Choice in the foundation of logic. J. Symb. Log. 50, 412–422 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Granas, C.D. Horvath, On the order-theoretic Cantor theorem. Taiwan. J. Math. 4, 203–213 (2000)MathSciNetzbMATHGoogle Scholar
  24. 24.
    P.R. Halmos, Measure Theory (Springer, New York, 1974)zbMATHGoogle Scholar
  25. 25.
    A. Hamel, Equivalents to Ekeland’s variational principle in uniform spaces. Nonlinear Anal. 62, 913–924 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D.H. Hyers, G. Isac, T.M. Rassias, Topics in Nonlinear Analysis and Applications (World Scientific Publishing, Singapore, 1997)CrossRefzbMATHGoogle Scholar
  27. 27.
    G. Isac, On Pareto efficiency. A general constructive existence principle, in Combinatorial and Global Optimization, ed. by P.M. Pardalos et al. (World Scientific Publishing, Singapore, 2002), pp. 133–144CrossRefGoogle Scholar
  28. 28.
    G. Isac, C. Tammer, Nuclear and full nuclear cones in product spaces: Pareto efficiency and an Ekeland type variational principle. Positivity 14, 1–28 (2004)zbMATHGoogle Scholar
  29. 29.
    C.F.K. Jung, On generalized complete metric spaces. Bull. Am. Math. Soc. 75, 113–116 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 44, 381–391 (1996)MathSciNetzbMATHGoogle Scholar
  31. 31.
    B.G. Kang, S. Park, On generalized ordering principles in nonlinear analysis. Nonlinear Anal. 14, 159–165 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    S. Kasahara, On some generalizations of the Banach contraction theorem. Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 427–437 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    W.A.J. Luxemburg, On the convergence of successive approximations in the theory of ordinary differential equations (II). Ind. Math. 20, 540–546 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    G.H. Moore, Zermelo’s Axiom of Choice: Its Origin, Development and Influence (Springer, New York, 1982)CrossRefzbMATHGoogle Scholar
  35. 35.
    Y. Moskhovakis, Notes on Set Theory (Springer, New York, 2006)Google Scholar
  36. 36.
    L. Nachbin, Topology and Order (D. van Nostrand, Princeton, 1965)zbMATHGoogle Scholar
  37. 37.
    A.B. Németh, Between Pareto efficiency and Pareto ɛ-efficiency. Optimization 20, 615–637 (1989)Google Scholar
  38. 38.
    T. Precupanu, Linear Topological Spaces and Fundamentals of Convex Analysis (Romanian) (Editura Academiei Române, Bucureşti, 1992)zbMATHGoogle Scholar
  39. 39.
    E. Schechter, Handbook of Analysis and its Foundation (Academic Press, New York, 1997)zbMATHGoogle Scholar
  40. 40.
    A. Szaz, An improved Altman type generalization of the Brezis Browder ordering principle. Math. Commun. 12, 155–161 (2007)MathSciNetzbMATHGoogle Scholar
  41. 41.
    A. Tarski, Axiomatic and algebraic aspects of two theorems on sums of cardinals. Fund. Math. 35, 79–104 (1948)MathSciNetzbMATHGoogle Scholar
  42. 42.
    D. Tataru, Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 163, 345–392 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    M. Turinici, A generalization of Brezis-Browder’s ordering principle. An. Şt. Univ. “A. I. Cuza” Iaşi (S I-a: Mat) 28, 11–16 (1982)Google Scholar
  44. 44.
    M. Turinici, Metric variants of the Brezis-Browder ordering principle. Demonstratio Math. 22, 213–228 (1989)MathSciNetzbMATHGoogle Scholar
  45. 45.
    M. Turinici, A monotone version of the variational Ekeland’s principle. An. Şt. Univ. “A. I. Cuza” Iaşi (S. I-a: Mat) 36, 329–352 (1990)Google Scholar
  46. 46.
    M. Turinici, Minimal points in product spaces. An. Şt. Univ. “Ovidius” Constanţa (Ser. Math.) 10, 109–122 (2002)Google Scholar
  47. 47.
    M. Turinici, Projective maximal principles in general vector spaces. Libertas Math. 29, 25–36 (2008)MathSciNetzbMATHGoogle Scholar
  48. 48.
    M. Turinici, Brezis-Browder principle and Dependent Choice. An Şt. Univ. “Al. I. Cuza” Iaşi (Mat.) 57, 263–277 (2011)Google Scholar
  49. 49.
    S. Turinici, M. Turinici, Projective metrics on abstract ordered sets. Mathematica (Cluj) 34 (57), 81–88 (1992)Google Scholar
  50. 50.
    E.S. Wolk, On the principle of dependent choices and some forms of Zorn’s lemma. Can. Math. Bull. 26, 365–367 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    J. Zhu, S.J. Li, Generalization of ordering principles and applications. J. Optim. Theory Appl. 132, 493–507 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.“A. Myller” Mathematical Seminar“A. I. Cuza” UniversityIaşiRomania

Personalised recommendations