Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A GENERIC RESULT IN VECTOR OPTIMIZATION | A GENERIC RESULT IN VECTOR OPTIMIZATION ALEXANDER J. ZASLAVSKI Received 17 November 2005 Revised 19 March 2006 Accepted 24 March 2006 We study a class of vector minimization problems on a complete metric space such that all its bounded closed subsets are compact. We show that for most in the sense of Baire category problems in the class the sets of minimal values are infinite. Copyright 2006 Alexander J. Zaslavski. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The study of vector optimization problems has recently been a rapidly growing area of research. See for example 1-5 and the references mentioned therein. In this paper we study a class of vector minimization problems on a complete metric space such that all its bounded closed subsets are compact. This class of problems is associated with a complete metric space of continuous vector functions defined below. For each F from we denote by v F the set of all minimal elements of the image F X F x x e X . In this paper we will study the sets v F with F e . It is clear that for a minimization problem with only one criterion the set of minimal values is a singleton. In the present paper we will show that for most F e in the sense of Baire category the sets v F are infinite. Such approach is often used in many situations when a certain property is studied for the whole space rather than for a single element of the space. See for example 7 8 and the references mentioned there. Our results show that in general the sets v F F e are rather complicated. Note that in our paper as in many other works on optimization theory 1-6 inequalities are of great use. In this paper we use the convention that oo to 1 and denote by Card E the cardinality of the set E. Let R be the set of real numbers and let n be a natural number. Consider the finitedimensional
