Minimax Theorems for ϕ-convex Functions with Applications

• Autorzy: Ewa M. Bednarczuk , Monika Syga
• Języki publikacji: EN

Abstrakty

EN

We investigate minimax theorems for ϕ-convex functions. As an application we provide a formula for the ϕ- conjugation of the pointwise maximum of ϕ- convex functions. (original abstract)

Słowa kluczowe

PL

Modele optymalizacyjne; Teoria optymalizacji; Programowanie matematyczne

EN

Optimizing models; Optimization theory; Mathematical programming

• Czasopismo: Control and Cybernetics
• Rocznik: 2014
• Tom: 43
• Numer: nr 3
• Strony: 421-437

Twórcy

autor

Ewa M. Bednarczuk

• Systems Research Institute of the Polish Academy of Sciences

autor

Monika Syga

• Warsaw University of Technology

Bibliografia

• AUBIN, J.P. (1998), Optima and Equilibria. Springer, New York-Berlin- Heidelberg.
• BOT˛, R.I., WANKA, G. (2008), The conjugate of the pointwise maximum of two convex functions revisited. J. Glob. Optim. 41, 625-632.
• BURACHIK, R.S., JEYAKUMAR, V. (2005), A new geometric condition for Fenchel's duality in infinite dimensions. Math. Program. 104B, 229-233.
• BURACHIK, R. S., RUBINOV, A. (2008), On abstract convexity and set valued analysis. J. Nonlinear and Convex Analysis 9, 105-123.
• DOLECKI, S., KURCYUSZ, S. (1978), On ϕ-convexity in extremal problems. SIAM J. Control and Optimization 16, 277-300.
• EKELAND, I., TEMAM, R. (1976), Convex Analysis and Variational Problems. North-Holland, Amsterdam.
• FAN, K. (1953), Minimax theorems. Proc. Nat. Acad. Sci. 39, 42-47.
• FITZPATRICK, S.P., SIMONS, S. (2000), On the pointwise maximum of convex functions. Proc. Am. Math. Soc. 128, 3553-3561.
• KINDLER, J. (1990), On a minimax theorem of Terkelsen's. Arch. Math. 55, 573-583.
• KINDLER, J., TROST, R. (1989), Minimax theorems for interval spaces. Act. Mat. Hung. 54, 39-49.
• JEYAKUMAR, V., RUBINOV, A.M., WU, Z.Y. (2007), Generalized Fenchel's conjugation formulas and duality for abstract convex functions. J. Optim. Theory Appl. 132, 441-458.
• NAGATA, J.I. (1985), Modern General Topology. North-Holland, Amsterdam.
• 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.
• PALLASCHKE, D., ROLEWICZ, S. (1997), Foundations of Mathematical Optimization. Kluwer Academic, Dordecht.
• RICCERI, B. (1993), Some topological mini-max theorems via an alternative principle for multifunctions. Arch. Math. 60, 367-377.
• RICCERI, B. (2008), Recent Advances in Minimax Theory and Applications. Pareto Optimality, Game Theory And Equilibria. Springer Optimization and Its Applications 17, 23-52.
• ROLEWICZ, S. (2003), ϕ-convex functions defined on a metric spaces. J. Math. Sci. 115, 2631 2652.
• RUBINOV, A.M. (2000), Abstract Convexity and Global Optimization. Kluwer Academic, Dordrecht.
• SIMONS, S. (1972), Maximinimax, minimax and antiminimax theorems and a result of R.C. James. Pacific Journal of Mathematics 40 (3), 709-718.
• SIMONS, S. (1990), On Terkelson's minimax theorem. Bull. Inst. Math. Acad. Sinica 18, 35-39.
• SIMONS, S. (1994), A flexible minimax theorem. Acta Math. Hungar. 63, 119- 132.
• 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.
• SION, M. (1958), On general minimax theorems. Pac. J. Math. 8, 171-176.
• SINGER, I. (1997), Abstract Convex Analysis. Wiley-Interscience, New York.
• SOLTAN, V.P. (1984), Introduction to Axiomatic Theory of Convexity (in Russian). Śtiinca, Kishiniev.
• STEFANESCU, A. (1985), A general min-max theorem. Optimization 16, 405- 413.
• STEFANESCU, A. (2007), The minimax equality; sufficient and necessary conditions. Acta Math. Sinica, English Series 23, 677-684.
• TERKELSEN, F. (1972), Some Minimax Theorems. Math. Scand. 31, 405-413.
• TUY, I. (1974), On a general minimax theorem. Soviet Math. Dokl. 15, 1689- 1693.
• WU, W.-T. (1959), A remark on the fundamental theorem in the theory of games. Sci. Rec. New Set. 3, 229-233.