chapter  2
34 Pages

Optimality and Duality

We say that f is a (real-valued) function on Rn, denoted f : Rn → R, if a real value f(x) is associated to any x ∈ Rn. A function is also called a mapping (or map). When dealing with optimization problems, it is often convenient to consider a function f while allowing to take +∞ as the value of f(x). In such a case, f is called an extended real valued function, and we write f : Rn → R ∪ {+∞} (or, f : Rn → (−∞,+∞]). In what follows, we use V to denote a Euclidean space, as done in Chapter 1, although we may simply consider V = Rn without loss of generality.