Optimization problems; Introduction

Abstract

If a problem in the real world is described (and thus necessarily idealized) by a mathematical model, then the problem often calls for maximizing, or minimizing, some function of the variables which describe the problem. For example, it may be required to calculate the conditions of operation of an industrial process which give the maximum output, or quality, or which give the minimum cost. Such calculations differ from the classical ‘maximum-minimum’ problems of the calculus textbooks, because the variables of the problem are nearly always subject to restrictions — equations or inequalities — called constraints. It is therefore not enough to equate a derivative, or gradient, to zero to find a maximum or minimum.