# 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:

• In equation solving there must be exactly as many equations as unknowns.
• In optimisation there are always a different number of unknowns and equations.
• Normally there are more unknowns than equations. the procedure thus involves choosing values of the extra `free' or `design' variables in order to either maximise or minimise the value of an objective function.
• The objective function (o.f.) can be function of the problem and design variables, but it must be a single valued function. It is not possible to optimise with respect to more than a single criterion.
• Mathematically maximising an o.f. is exactly the same as minimising minus the o.f.
• A special case of also optimisation occurs when there are are fewer unknowns than equations. This includes the problem of data fitting and is discussed in section 5.2.1

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.'

## 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.