We investigate a new approach to optimization with respect to set-valued objective functions. The basic idea is to understand the set-valued objective map as a function into a hyperspace, in fact we consider functions with values in the space C of all closed convex subsets of finite dimensional normed vector space. This space is provided with an appropriate ordering relation. It turns out that the considerations can be reduced to the case of the usual set inclusion. Chapter 1 is devoted to the study of algebraic and order theoretical properties of C. This involves an investigation about the possibility to embed certain classes of C into a linear space. This results in an extension of the classical embedding assertions of Radström from 1952. The second chapter deals with topological properties of C. We introduce a convergence class in C, which seems to be new. We define this convergence by appropriate concepts of upper and lower limits. A characterization by (a certain type of) scalar convergence is indicated. We compare this convergence with well-established concepts. In Chapter 3, we investigate functions with values in C. Based on the results of the second chapter, we develop semi-continuity concepts. Moreover, we introduce the notion of a conjugate of a C-valued function. The main result of this section is a biconjugation theorem for C-valued functions. Chapter 4 is devoted to optimization problems with C-valued objective function. In particular, we draw our attention to duality theory. We prove weak as well as strong duality assertions based on Fenchel as well as Lagrange approach. The biconjugation theorem, established in Chapter 3, is used as a main tool for the proof of the strong duality assertions. The last chapter contains a comparison of the duality results for optimization with set inclusion and duality assertions in vector optimization.