2.5 A Final Example

As a third example, consider the constrained problem in the variables
, and y

and the simple bounds

As before, there are a number of ways of casting this problem in the form (2.1)-(2.4). We chose to decompose the problem as follows:

- the objective function comprises two groups, the first of which
uses the non-trivial group function
. This group contains a single
*linear*element; the element function is y. There is also a nonlinear element . This element function has three elemental variables, , and , say (with , and ); there is a useful transformation from elemental to internal variables of the form and and the element function may then be represented as . The second group may be considered as a quadratic objective group, and written as , where and . - The next set of groups,
inequality constraints,
are of the form (2.4) with no lower
bounds.
Each uses
the trivial group
function
and contains a
single
*linear*element, , and two*nonlinear*elements and . Both nonlinear elements are of the same type, , for appropriate variables and and parameter , and there is no useful transformation to internal variables. - The following set of groups,
again inequality constraints,
for
, are of
the form (2.4) with both lower and upper bounds.
Each uses the
non-trivial group function
and contains a
single
*nonlinear*element of the type for an appropriate variable . Notice that the group types for these groups and for the objective function group are both of the form , for some parameter , and it may prove more convenient to use this form to cover both sets of groups. - The last group,
an equality constraint,
,
is of the form (2.3).
Again, this group uses the trivial group function
and contains a single
*linear*element, , and a single*nonlinear*element of the type for appropriate elemental variables and . Once more, a single internal variable, can be used and the element is then represented as .

Thus we see that we can consider our problem to be made up of 201 groups of two different types as well as an quadratic objective group so we will have to provide our optimization procedure with function and derivative values for these at some stage. There are 200 nonlinear elements of four different types and again this means that we shall have to provide function and derivative values for these. As for the previous example, there is so much structure to this problem that it would be inefficient to pass the data group-by-group and element-by-element. Again, we will introduce ways to specify this repetitious structure using a convenient shorthand.