Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

30 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
bought from, or sold to rms outside the state, statistics on which are not available.
De ne
yt =number (in millions) of growing pullets in the state, on the rst day of
month t.
dt =number (in millions) of day-old-chickens hatched by hatcheries in the
state in month t (from government statistics).
Here dt are not variables, but are the given data. People in the business of producing
chicken feed are very much interested in getting estimates of yt from dt. This provides
useful information to them in their production planning, etc. Not all the day-oldchickens
placed by hatcheries in a month may be alive in a future month. Also, after
ve months of age, they are recorded as hens and do not form part of the population of
growing pullets. So the appropriate linear regression model for yt as a function of the
dt's seems to be yt = 0 + P5 i=1 idt;i, where 0 is the number of pullets in census,
which are not registered as being hatched (pullets imported into the State, or chickens
exported from the State), and i is a survival rate (the proportion of chickens placed in
month t;i that are alive in month t, i = 1 to 5). We, of course, expect the parameters
i to satisfy the constraints
0 < = 5 < = 4 < = 3 < = 2 < = 1 < =
1 : (1:21)
To get the best estimates for the parameters = ( 0 1 2 3 4 5) T from past
data, the least squares method could be used. Given data on yt, dt over a period of
time (say for the last 10 years), de ne L2( )= P t (yt ; 0 ; P5 i=1 idt;i) 2 . Under the
least squares method the best values for are taken to be those that minimize L2( )
subject to the constraints (1.21). This is clearly a quadratic programming problem.
One may be tempted to simplify this problem by ignoring the constraints (1.21)
on the parameters . The unconstrained minimum of L2( ) can be found very easily
by solving the system of equations @L2( )
=0.
@
There are two main di culties with this approach. The rst is that the solution of
this system of equations requires the handling of a square matrix (aij) with aij =1=(i+
j ; 1), known as the Hilbert matrix, which is di cult to use in actual computation
because of ill-conditioning. It magni es the uncertainty in the data by very large
factors. We will illustrate this using the Hilbert matrix of order 2. This matrix is
8
H2 = >: 1 1
3
Consider the following system of linear equations with H2 as the coe cient matrix.
x1
1
1
2
1
2
x2
1
2
1
3
1
3
9
> :
b1
b2