A Generic Result in Vector Optimization

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 ∈ X}. In this paper, we will study the sets v(F) with F ∈ . 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 ∈ (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 ∈ , 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 ∞/∞= 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 space Rn with the Chebyshev norm