Using the Embedding Theorem to Solve Interval-Valued Optimization Problems

The space of all bounded closed intervals cannot form a vector space because the concept of an additive inverse cannot be considered. Therefore, this paper presents an embedding theorem to show that the space of all bounded closed intervals can be embedded into a Banach space. In this case, the partial orders among the space of all bounded closed intervals can be proposed via the embedding theorem by considering the ordering cones in that Banach space. After a partial order is introduced in the interval-valued optimization problem, the solution concepts of interval-valued optimization problem can be naturally defined. On the other hand, using the embedding theorem, an auxiliary vector optimization problem is introduced such that solving the original interval-valued optimization problem is equivalent to solving the auxiliary vector optimization problem. The technique of scalarization is proposed to solve the auxiliary vector optimization problem. Three practical ordering cones are considered to study the linear type of interval-valued optimization problem for the purpose of presenting practical applications.