Abstract
In the present paper, the notions of cones of the same sense and cones of the opposite sense are introduced. The basic properties of these cones are examined. The most essential property is the following: ifX is a Banach space, then, in the dual spaceX*, the algebraic sum of a finite number of weakly * closed convex cones of the same sense is weakly * closed. On the basis of the properties of these cones, some necessary conditions for the existence of extrema in extremal problems are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dubovitskii, A. Ya., andMilyutin, A. A.,Extremum Problems in the Presence of Constraints, Żurnal Vycislitel'nei Matematiki i Matematiceskoi Fiziki, Vol. 5, pp. 395–453, 1965.
Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.
Hermes, H., andLasalle, J. P.,Functional Analysis and Time Optimal Control, Academic Press, New York, New York, 1969.
Rolewicz, S.,--Analiza Funkcjonalna i Teoria Sterowania, WNT, Warsaw, Poland, 1977 (in Polish).
Ioffe, A. D., andTikhomirov, V. M.,Theory of Extremal Problems, Academic Press, New York, New York, 1979.
Girsanov, I. V.,Lectures on Mathematical Theory of Extremum Problems, Springer-Verlag, New York, New York, 1972.
Walczak, S.,On Some Control Problem (to appear).
Walczak, S.,The Local Extremum Principle for Problems with Constraints upon Phase Coordinates (to appear).
Dunford, N., andSchwartz, J.,Linear Operators, John Wiley and Sons (Interscience Publishers), New York, New York, 1958.
Author information
Authors and Affiliations
Additional information
Communicated by D. G. Luenberger
Rights and permissions
About this article
Cite this article
Walczak, S. Some properties of cones in normed spaces and their application to investigating extremal problems. J Optim Theory Appl 42, 561–582 (1984). https://doi.org/10.1007/BF00934567
Issue Date:
DOI: https://doi.org/10.1007/BF00934567