Section 5.2: Overview and General Notes

In equation solving our objective is to find the answer to a problem. There is in general just one valid answer (subject to the proviso that formally a given, well posed set of equations may occasionally have more that one).

In optimisation we are in a situation where a variety of acceptable answers are available to our problem. The aim is to determine which one is best in terms of some specified criterion.

Note that:


The notes below cover the following topics:

General Notation for Optimisation Problems

The simplest general form of an optimisation problem may be stated in words as:

`Given a set of variables xi, i = 1, 2 ,... n and an objective function P (x1, x2, x3 ...) find values for the xi such that P is a minimum (or a maximum).'

This is called an unconstrained optimisation problem.

The next most complicated class of problem conceptually is an optimisation with inequality constraints. We may write the definition of this as:

`Given a set of variables xi, i = 1, 2 ,... n and an objective function P (x1, x2, x3 ...) find values for the xi such that P is a minimum but such that a set of inequalities:
g1 (x1, x2, x3 ...) < 0
g2 (x1, x2, x3 ...) < 0
.....
gm (x1, x2, x3 ...) < 0
are not violated.'

As noted above we cam maximise the o.f. by changing its sign and minimising, and so can drop the explicit reference to maximisation. Similarly, we can express any kind of inequality in the above form by appropriate manipulation of the expression. The above definitions are of an optimisation involving n decision variables. Note that the number of inequality constraints, given above as m is in no way connected to the number of variables n. There may be either more or fewer inequality constraints than variables.

One Variable Optimisation

When n = 1 it is particularly easy to visualise optimisation, and a number of special methods are available for the above two types of problem.