# Section 5.4.1: Single Variable NLP Optimisation Problems

Single variable problems can be dived into those which can
conveniently be solved analytically and those which it is best
to solve by numerical search.
Problems which appear to be multivariable but include * equality * constraints may sometimes be reduced to single variable form.

## Example 1

A rectangular platinum catalyst gauze is to provide 300cm¼ of effective
area, but requires installation margins of 4cm on each of its longer sides and
6cm on the shorter sides.
Find the dimension which minimises total area.

Let the overall dimensions of the gauze be *x* cm by *y* cm.
The objective function to be minimised is then clearly:

* P = x y *

The mounting margins provide the active width of (*x*-8)
and height of (*y*-12), which must give a total exposed area of
300cm¼ so that there will be an equality constraint:

(*x*-8) (*y*-12) = 300

### Solution method

A solution in two variables is in principle possible, but since, as with
equation solving, single variable methods are generally much simpler and more
robust, we can use each equality constraint to * reduce the
dimension of the problem by one. *
By rearrangement of the equality constraint we can eliminate either of
the variable in the o.f. E.g. rearranging to get *x*:

*x* = (204+8*y*) / (4-12*y*)

The unconstrained single variable o.f. then is:

P = *x y * = * y * (204+8*y*) / (4-12*y*)

The minimum can be found either by differentiating and setting equal to
zero or by a direct search.

## Example 2

The expression for work per unit volume
of gas in a 2-stage compressor with intercooling
going from pressure P1 to P2 then to P3 is:
W/V = (P1/K) [(P2/P1)^{K} - 2 + (P3/P2)^{K}]

Here K=(k-1)/k where k is the ration of heat capacities, approximately 1.4.

For the case where P1 = 1 atm and P3 = 4 atm confirm the well known result that
P2 should be 2 atm.

### Solution method

This is entirely straightforward since clearly W/V should be minimised, and
the single decision variable is P2. a numerical search between 1 and 4 atm
is most convenient.

Next - Section 5.4.2: LP Problems

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