Introduction

Abstract

Non-linear optimization is concerned with the characterisation and location of maxima or minima of non-linear functions. Such problems are widespread in the mathematical modelling of real world systems from a very broad range of applications. To introduce the idea, consider a simple problem (Rosenbrock, 1960). It is proposed to send a rectangular parcel, but the firm of carriers restricts the dimension in any direction to a maximum of 42 inches and the girth to a maximum of 72 inches. Which dimensions give the greatest volume? We essentially need to calculate the maximum value of the function x₁x₂x₃, where x₁, let us say, represents the length, x₂ the width and x₃ the depth of the parcel. However, the possible values of x₁, x₂ and x₃ are restricted by the set of relations

$$\begin{array}{*{20}{l}} {0 \leqslant {x_1} \leqslant 42} \\ {0 \leqslant {x_2} \leqslant 42} \\ {0 \leqslant {x_3} \leqslant 42} \\ {0 \leqslant {x_1} + 2{x_2} + 2{x_3} \leqslant 72} \end{array}$$

which are known as constraints. This is an example of a constrained non-linear optimization problem (non-linear because the function to be maximized is non-linear, even though the constraint functions are linear) with three variables. Many problems do not have constraints.