Appl Math Optim 36:313–322 (1997)
1997 Springer-Verlag New York Inc.
Generalized Semicontinuity and Existence Theorems for
Cone Saddle Points
Department of Information Science, Faculty of Science,
Hirosaki University, Hirosaki 036, Japan
Abstract. This paper is concerned with existence theorems for generalized saddle
points (cone saddle points) of vector-valued functions. A concept of lower semicon-
tinuity for vector-valued functions is introduced and its properties are investigated.
Previous results of the author are extended by using the lower semicontinuity.
Key Words. Vector optimization, Cone-continuity, Cone saddle points, Minimax
AMS Classiﬁcation. Primary 90C29, Secondary 49J35.
It is well known that the convexity and continuity of real-valued functions play very
important roles in the area of nonlinear optimization as well as in various ﬁelds of
mathematics. In particular, convex programming, separation theorem, minimax theorem,
and saddle point problem are closely connected with them. A great deal of work related
to such properties has been given, and various relaxations and modiﬁcations of the
properties have also been investigated to extend optimal conditions and so on; see  and
. On the other hand, some of those properties have been generalized to vector-valued
functions to explore and characterize several optimal solutions, in particular, efﬁcient
solutions in the areaof vectoroptimization. In  and some types of cone-convexity
and continuity are introduced, and then vector-valued minimax theorems are proved for
vector-valued functions which satisfy these properties.
This work was based on research 07740136 supported by a Grant-in-Aid for Scientiﬁc Research (Grant-
in-Aid for Encouragement of Young Scientists) from the Ministry of Education, Science, and Culture of Japan.