Reference Work Entry

Encyclopedia of Operations Research and Management Science

pp 650-658

Date:

Global Optimization

  • Hoang TuyAffiliated withInstitute of Mathematics, Vietnam Academy of Science and Technology Email author 
  • , Steffen RebennackAffiliated withDivision of Economics & Business, Colorado School of Mines
  • , Panos M. PardalosAffiliated withDepartment of Industrial and Systems Engineering, University of Florida

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 sol ...

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