−C Representing Functionals

This chapter provides an introduction to order analysis. Based on a certain property of Bishop-Phelps, penalty, and scalarizing functionals, we introduce functionals representing the negative preorder cone. This representation is done in such a way that the negative preorder cone equals the sublevel set of such a functional with respect to the level 0. The preorder can then be treated using nonlinear analysis, and this is the key to order analysis. We investigate various properties of these \(-C\) representing functionals like basic properties, sublinearity, directional differentiability, subdifferential, zeros, and separation. Furthermore, we extend this approach to certain set-valued maps so that variable order structures can also be studied.