Problem 9.7

©2003 T. D, Morley
School of Math
Georgia Institute of Technology

In[2]:=

(A = {{2, 6, -1}, {4, -4, 3}, {3, 1, -4}, {7, -3, -6}, {1, -5, 2}, {7, 13, -6}, {-6, -2, 3}}) // MatrixForm

Out[2]//MatrixForm=

( 2 6 -1 ) 4 -4 3 3 1 -4 7 -3 -6 1 -5 2 7 13 -6 -6 -2 3

In[3]:=

(b = {-6, 0, 3, 10, 2, -7, 0}) // MatrixForm

Out[3]//MatrixForm=

( -6 ) 0 3 10 2 -7 0

In[4]:=

constraints = Thread[A . {x, y, z} <= b]

Out[4]=

{2 x + 6 y - z <= -6, 4 x - 4 y + 3 z <= 0, 3 x + y - 4 z <= 3, 7 x - 3 y - 6 z <= 10, x - 5 y + 2 z <= 2, 7 x + 13 y - 6 z <= -7, -6 x - 2 y + 3 z <= 0}

In[8]:=

ConstrainedMax[0, constraints, {x, y, z}]

ConstrainedMax :: lpsnf : No solution can be found that satisfies the constraints.

Out[8]=

ConstrainedMax[0, {2 x + 6 y - z <= -6, 4 x - 4 y + 3 z <= 0, 3 x + y - 4 z <= 3, 7 x - 3 y - 6 z <= 10, x - 5 y + 2 z <= 2, 7 x + 13 y - 6 z <= -7, -6 x - 2 y + 3 z <= 0}, {x, y, z}]

Ok  -- Look at dual

In[7]:=

vars = Table[z[i], {i, 1, 7}]

Out[7]=

{z[1], z[2], z[3], z[4], z[5], z[6], z[7]}

In[9]:=

dualConstraints = Thread[Transpose[A] . vars == 0]

Out[9]=

{2 z[1] + 4 z[2] + 3 z[3] + 7 z[4] + z[5] + 7 z[6] - 6 z[7] == 0, 6 z[1] - 4 z[2] + z[3] - 3 z[4] - 5 z[5] + 13 z[6] - 2 z[7] == 0, -z[1] + 3 z[2] - 4 z[3] - 6 z[4] + 2 z[5] - 6 z[6] + 3 z[7] == 0}

In[11]:=

ConstrainedMin[b . vars, dualConstraints, vars]

ConstrainedMin :: lpsub : The problem is unbounded.

Out[11]=

{-∞, {z[1] -> Indeterminate, z[2] -> Indeterminate, z[3] -> Indeterminate, z[4] -> Indeterminate, z[5] -> Indeterminate, z[6] -> Indeterminate, z[7] -> Indeterminate}}

Looks like  the primal is Inconsistant, and the dual is unbounded. Just as it should.

Converted by Mathematica  (March 25, 2003)