Global Optimization

Introduction

Consider an optimization problem of the general form

$$ \text{min}\left\{ {f(x)|\:{g_i}(x) \le 0, i = 1, \ldots, m,\;x \in X} \right\}\quad {\text{(P)}} $$

where $X$ is a closed convex set in ${\mathbb{\mathbb R}^n},$$f:\Omega \to \mathbb{\mathbb R},$ and ${g_i}:\Omega \to \mathbb{\mathbb R},\:i = 1, \ldots, m,$ are continuous functions defined on some open set $\Omega$ in ${\mathbb{\mathbb R}^n}$ containing $X.$ Setting

$$ D = \left\{ {x \in X|\:{g_i}(x) \le 0,\:i = 1, \ldots, m} \right\}, $$

the problem can also be written as

$$ \text{min}\left\{ {f(x)|\:x \in D} \right\}. $$

Any point $\bar{x} \in D$ is called a feasible solution of the problem. A feasible solution $\bar{x}$ is called a global optimal solution if it is the best of all feasible solutions, i.e., if it satisfies

$$ f(\bar{x}) \le f(x)\quad \forall x \in D. $$

(1)

A feasible solution $\bar{x}$is called a local optimal solution if it is the best among all feasible...