As a third example, consider the constrained problem in the variables
, 
and y
subject to the constraints
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 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 
.
are of the form (2.4) with no lower
bounds.
Each uses
the trivial group
function 
and contains a
single linear element,
y, 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.
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.
,
is of the form (2.3).
Again, this group uses the trivial group function
and contains a single linear
element,
-1, 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 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.
