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Minimax theorems for ϕ−convex functions with applications

Treść / Zawartość
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We investigateminimax theorems for ϕ−convex functions. As an application we provide a formula for the ϕ- conjugation of the pointwise maximum of ϕ- convex functions.
Rocznik
Strony
421--437
Opis fizyczny
Bibliogr. 30 poz., rys.
Twórcy
  • Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
autor
  • Warsaw University of Technology, Faculty of Mathematics and Information Science, Koszykowa 75, 00-662 Warsaw, Poland
Bibliografia
  • 1. AUBIN, J.P. (1998) Optima and Equilibria. Springer, New York–Berlin– Heidelberg.
  • 2. BOT˛, R.I. and WANKA, G. (2008) The conjugate of the pointwise maximum of two convex functions revisited. J. Glob. Optim. 41, 625–632.
  • 3. BURACHIK, R.S. and JEYAKUMAR, V. (2005) A new geometric condition for Fenchel’s duality in infinite dimensions. Math. Program. 104B, 229–233.
  • 4. BURACHIK, R. S. and RUBINOV, A. (2008) On abstract convexity and set valued analysis. J. Nonlinear and Convex Analysis 9, 105–123.
  • 5. DOLECKI, S. and KURCYUSZ, S. (1978) On ϕ−convexity in extremal problems. SIAM J. Control and Optimization 16, 277–300.
  • 6. EKELAND, I. and TEMAM, R. (1976) Convex Analysis and Variational Problems. North–Holland, Amsterdam.
  • 7. FAN, K. (1953) Minimax theorems. Proc. Nat. Acad. Sci. 39, 42–47.
  • 8. FITZPATRICK, S.P. and SIMONS, S. (2000) On the pointwise maximum of convex functions. Proc. Am. Math. Soc. 128, 3553–3561.
  • 9. KINDLER, J. (1990) On a minimax theorem of Terkelsen’s. Arch. Math. 55, 573–583.
  • 10. KINDLER, J. and TROST, R. (1989) Minimax theorems for interval spaces. Act. Mat. Hung. 54, 39–49.
  • 11. JEYAKUMAR, V., RUBINOV, A.M. and WU, Z.Y. (2007) Generalized Fenchel’s conjugation formulas and duality for abstract convex functions. J. Optim. Theory Appl. 132, 441–458.
  • 12. NAGATA, J.I. (1985) Modern General Topology. North–Holland, Amsterdam.
  • 13. VON NEUMANN, J. (1928) Zur Theorie der Gesellschaftspiele. Math. Ann. 100, 295–320. For an English translation see: On the theory of games of strategy. Contributions to the theory of games 4, Princeton. Univ. Press (1959), 13–42.
  • 14. PALLASCHKE, D. and ROLEWICZ, S. (1997) Foundations of Mathematical Optimization. Kluwer Academic, Dordecht.
  • 15. RICCERI, B. (1993) Some topological mini-max theorems via an alternative principle for multifunctions. Arch. Math. 60, 367–377.
  • 16. RICCERI, B. (2008) Recent Advances in Minimax Theory and Applications. Pareto Optimality, Game Theory And Equilibria. Springer Optimization and Its Applications 17, 23–52.
  • 17. ROLEWICZ, S. (2003) _−convex functions defined on a metric spaces. J. Math. Sci. 115, 2631 2652.
  • 18. RUBINOV, A.M. (2000) Abstract Convexity and Global Optimization. Kluwer Academic, Dordrecht.
  • 19. SIMONS, S. (1972) Maximinimax, minimax and antiminimax theorems and a result of R.C. James. Pacific Journal of Mathematics 40 (3), 709–718.
  • 20. SIMONS, S. (1990) On Terkelson’s minimax theorem. Bull. Inst. Math. Acad. Sinica 18, 35–39.
  • 21. SIMONS, S. (1994) A flexible minimax theorem. Acta Math. Hungar. 63, 119– 132.
  • 22. SIMONS, S. (1995) Minimax theorems and their proofs. In: D.-Z. Du and P. M. Pardalos, eds., Minimax and Applications. Kluwer Academic Publishers, 1–23.
  • 23. SION, M. (1958) On general minimax theorems. Pac. J. Math. 8, 171–176.
  • 24. SINGER, I. (1997) Abstract Convex Analysis. Wiley-Interscience, New York.
  • 25. SOLTAN, V.P. (1984) Introduction to Axiomatic Theory of Convexity (in Russian). Śtiinca, Kishiniev.
  • 26. STEFANESCU, A. (1985) A general min-max theorem. Optimization 16, 405– 413.
  • 27. STEFANESCU, A. (2007) The minimax equality; sufficient and necessary conditions. Acta Math. Sinica, English Series 23, 677–684.
  • 28. TERKELSEN, F. (1972) Some Minimax Theorems. Math. Scand. 31, 405–413.
  • 29. TUY, I. (1974) On a general minimax theorem. Soviet Math. Dokl. 15, 1689– 1693.
  • 30. WU,W.-T. (1959) A remark on the fundamental theorem in the theory of games. Sci. Rec. New Set. 3, 229–233.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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