Numerical Methods and Optimization

166 Numerical Methods and Optimization: An Introduction

X

y



d



h



ε



x



y



d



h



ε



x



FIGURE 8.3: Examples of interior points (x



and x



), boundary points (y



and y



), feasible directions (d



and d



), and infeasible directions (h



and h



).

of interior and boundary points, as well as feasible and infeasible directions

at the boundary points.

8.3 Local and Global Optimality

We consider a minimization problem in the form

minimize f(x)

subject to x ∈ X,

(8.7)

where f :IR

n

→ IR is an arbitrary function of n variables x

j

,j =1,...,n.

To solve this problem, one needs to find a feasible solution x

∗

that mini-

mizes f over X. Such a solution is called a global optimal ...