Abstract
In this work, as usual in vector-valued optimization, we consider the partial ordering induced in a topological vector space by a closed and convex cone. In this way, we define maximal and minimal sets of a vector-valued function and consider minimax problems in this setting. Under suitable hypotheses (continuity, compactness, and special types of convexity), we prove that, for every
there exists
such that β ≤ α (the exact meanings of the symbols are given in Section 2).
Similar content being viewed by others
References
Fan, K.,Minimax Theorems, Proceedings of the National Academy of Sciences of the USA, Vol. 39, pp. 42–47, 1953.
Moreau, J. J.,Théoremes “inf-sup”, Comptes Rendus des Séances de l'Academie des Sciences de Paris, Série I, Vol. 258, pp. 2720–2722, 1964.
Brezis, H., Niremberg, L., andStampacchia, G.,A Remark on Ky Fan's Minimax Principle, Bollettino dell'Unione Matematica Italiana, Series 4, Vol. 6, pp. 293–300, 1972.
Stachò, L. L.,Minimax Theorems Beyond Topological Vector Spaces, Acta Scientiarum Mathematicarum, Szeged, Vol. 42, pp. 157–164, 1980.
Bennati, M. L., andFerro, F.,Teoremi di Minimax, Report No. 113, Istituto per la Matematica Applicata del CNR, Genova, Italy, 1981.
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.
Tanino, T., andSawaragi, Y.,Duality Theory in Multiobjective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–529, 1979.
Corley, H. W.,Duality Theory for Maximizations with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 84, pp. 560–568, 1981.
Chew, K. L.,Maximal Points with Respect to Cone Dominance in Banach Spaces and Their Existence, Journal of Optimization Theory and Applications, Vol. 44, pp. 1–53, 1984.
Ferro, F.,Minimax Type Theorems for n-Valued Functions, Annali di Matematica Pura ed Applicata, Vol. 32, pp. 113–130, 1982.
Nieuwenhuis, J. W.,Some Minimax Theorems in Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 40, pp. 463–475, 1983.
Corley, H. W.,Games with Vector Payoffs, Journal of Optimization Theory and Applications, Vol. 47, pp. 491–498, 1985.
Borwein, J.,On the Existence of Pareto Efficient Points, Mathematics of Operations Research, Vol. 8, pp. 64–73, 1983.
Hartley, R.,On Cone Efficiency, Cone Convexity, and Cone Compactness, SIAM Journal on Applied Mathematics, Vol. 34, pp. 211–222, 1978.
Corley, H. W.,An Existence Result for Maximizations with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 31, pp. 277–281, 1980.
Hildebrand, W.,Core and Equilibrium of a Large Economy, Princeton University Press, Princeton, New Jersey, 1974.
Aubin, J. P., andCellina, A.,Differential Inclusions, Springer-Verlag, Berlin, Germany, 1984.
Author information
Authors and Affiliations
Additional information
Communicated by P. L. Yu
Rights and permissions
About this article
Cite this article
Ferro, F. A minimax theorem for vector-valued functions. J Optim Theory Appl 60, 19–31 (1989). https://doi.org/10.1007/BF00938796
Issue Date:
DOI: https://doi.org/10.1007/BF00938796