The purpose of this chapter is to outline an overview of the basic criteria for characterizing the solution of static optimization problems. Its intent is neither rigor nor generality but a sufficient understanding for the logic leading to the conditions of optimization and a sufficient degree of practicality. An immediate issue is whether we are satisfied with relative optima or require global optima as the solution of our problems. Any further elaboration necessitates a definition of these two notions.

Definition: A point x* ∈ Rn is a relative (or local) maximum point of a function f defined over Rn if there exists an ε > 0 such that f(x*) ≥ f(x) for all x ∈ Rn and |x - x*| < ε.

A point x* ∈ Rn is a strict relative (or local) maximumpoint of a function f defined over Rn if there exists an ε > 0 such that f(x*) > f(x) for all x ∈ Rn and |x - x*| < ε.

In general, we would like to deal with solution points that correspond to global optima.

Definition: A point x* ∈ Rn is a global maximum point of a function f defined over Rn if f (x*) ≥ f (x) for all x ∈ Rn.

Apoint x* ∈ Rn is a strict global maximum point of a function f defined over Rn if f (x*) > f (x) for all x ∈ Rn.