Abstract
We establish the following theorems: (i) an existence theorem for weak type generalized saddle points; (ii) an existence theorem for strong type generalized saddle points; (iii) a generalized minimax theorem for a vector-valued function. These theorems are generalizations and extensions of the author's recent results. For such extensions, we propose new concepts of convexity and continuity of vector-valued functions, which are weaker than ordinary ones. Some of the proofs are based on a few key observations and also on the Browder coincidence theorem or the Tychonoff fixed-point theorem. Also, the minimax theorem follows from the existence theorem for weak type generalized saddle points. The main spaces with mathematical structures considered are real locally convex spaces and real ordered topological vector spaces.
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References
Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley, New York, New York, 1969.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Ekeland, I. andTemam, R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.
Bazaraa, M. S., andShetty, C. M.,Nonlinear Programming: Theory and Algorithms, John Wiley, New York, New York, 1979.
Tanaka, T.,Cone-Convexity of Vector-Valued Functions, Science Reports of Hirosaki University, Vol. 37, pp. 170–177, 1990.
Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.
Hartley, R.,On Cone-Efficiency, Cone-Convexity, and Cone-Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.
Tanino, T., andSawaragi, Y.,Duality Theory in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–529, 1979.
Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, Orlando, Florida, 1985.
Henig, M. I.,Existence and Characterization of Efficient Decisions with Respect to Cones, Mathematical Programming, Vol. 23, pp. 111–116, 1982.
Jahn, J.,Scalarization in Vector Optimization, Mathematical Programming, Vol. 29, pp. 203–218, (1984).
Karwat, A. S.,On Existence of Cone-Maximal Points in Real Topological Linear Spaces, Israel Journal of Mathematics, Vol. 54, pp. 33–41 1986.
Tanaka, T.,On Cone-Extreme Points in R n, Science Reports of Niigata University, Vol. 23, pp. 13–24, 1987.
Luc, D. T.,Connectedness of the Efficient Point Sets in Quasiconcave Vector Maximization, Journal of Optimization Theory and Applications, Vol. 122, pp. 346–354, 1987.
Luc, D. T.,An Existence Theorem in Vector Optimization, Mathematics of Operations Research, Vol. 14, pp. 693–699, 1989.
Nieuwenhuis, J. W.,Some Minimax Theorems in Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 40, pp. 463–475, 1983.
Ferro, F.,Minimax Type Theorems for n-Valued Functions, Annali di Matematica Pura ed Applicata, Vol. 32, pp. 113–130, 1982.
Ferro, F.,A Minimax Theorem for Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 60, pp. 19–31, 1989.
Ferro, F.,A Minimax Theorem for Vector-Valued Functions, Part 2, Journal of Optimization Theory and Applications, Vol. 68, pp. 35–48, 1991.
Tanaka, T.,Some Minimax Problems of Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 59, pp. 505–524, 1988.
Tanaka, T.,Existence Theorems for Cone Saddle Points of Vector-Valued Functions in Infinite-Dimensional Spaces, Journal of Optimization Theory and Applications, Vol. 62, pp. 127–138, 1989.
Tanaka, T.,A Characterization of Generalized Saddle Points for Vector-Valued Functions via Scalarization, Nihonkai Mathematical Journal, Vol. 1, pp. 209–227, 1990.
Tanaka, T.,Two Types of Minimax Theorems for Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 68, pp. 321–334, 1991.
Browder, F. E.,Coincidence Theorems, Minimax Theorems, and Variational Inequalities, Contemporary Mathematics, Vol. 26, pp. 67–80, 1984.
Simons, S.,Cyclical Coincidences of Multivalued Maps, Journal of the Mathematical Society of Japan, Vol. 38, pp. 515–525, 1986.
Tychonoff, A.,Ein Fixpunktsatz, Mathematicsche Annalen, Vol. 111, pp. 767–776, 1935.
Jameson, G.,Ordered Linear Spaces, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, Vol. 141, 1970.
Nash, J.,Noncooperative Games, Annals of Mathematics, Vol. 54, pp. 286–295, 1951.
Tardella, F.,On the Image of a Constrained Extremum Problem and Some Applications to the Existence of a Minimum, Journal of Optimization Theory and Applications, Vol. 60, pp. 93–104, 1989.
Holmes, R. B.,Geometric Functional Analysis and Its Applications, Springer-Verlag, New York, New York, 1975.
Ha, C. W.,Minimax and Fixed-Point Theorems, Mathematische Annalen, Vol. 248, pp. 73–77, 1980.
Aubin, J. P., andCellina, A.,Differential Inclusions, Springer-Verlag, Berlin, Germany, 1984.
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Communicated by P. L. Yu
This paper is dedicated to Professor Kensuke Tanaka on his sixtieth birthday.
This paper was written when the author was a visitor at the Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata, Japan. The author is indebted to Prof. K. Tanaka for suggesting this work.
The author is very grateful to Prof. P. L. Yu for his useful encouragement and suggestions and to the referees for their valuable suggestions and comments.
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Tanaka, T. Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. J Optim Theory Appl 81, 355–377 (1994). https://doi.org/10.1007/BF02191669
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DOI: https://doi.org/10.1007/BF02191669