Math 1b Practical
March 9, 2009
LinearProgramming[c,A,b] solves what I would call the
transpose-dual-canonical form LP problem -- it finds
a vector
x
that minimizes the value of
cx
subject to
Ax >= b
and
x >= 0 .
In In[1], I define
LP[c,A,b]
which solves the
canonical form LP problem:
maximize
cx
subject
to Ax <= b , x >= 0.
If
A
is as in Out[2], c = {1,1,1,1}
and
b = {11,8,9},
a vector
x_0
that maximizes
cx
in the canonical
form problem is given in Out[3] as
{2.7, 0, 1.1, 1.5} .
The value of
cx
is
5.3, as shown in Out[4].
A vector
y_0
that minimizes
yb
in the dual canonical
form problem is {0.1, 0.3, 0.2} .
The value of
yb
is
5.3.
This checks the Duality Theorem.
Let
A
be the upper-left 3x4 submatrix of Out[7] and
consider the two-person zero-sum game defined by
A .
An optimal strategy for player I is to pick row 1 with
probability 1/7 and row 2 with probability 6/7.
An
optimal trategy for player II is to pick column 2 with
probability 4/7 and column 4 with probability 3/7.
For these strategies, the expected win for Player I
is 25/7.

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