Date: Jan 19, 2013 9:46 PM
Author: analyst41@hotmail.com
Subject: Go from any vertex to any other vertex in one Simplex method pivot.

Consider the LP

Max Z = x1 + x2

such that

0<=x1 <=1
0<=x2 <= 1.

If you start from (Z,x1,x2) = (0,0,0) to go the optimum (Z,x1,x2) =
(2,1,1) you need two pivots, because you can't go 'through" the unit
square under the Simplex method.

If you do the variable transformation

   y1 = x1 +x2
   y2 = x1 ? x2

    x1 = (y1 + y2)/2
    x2 = (y1 ? y2)/2

The LP becomes

Max Z  = y1

Such that

0<= y1+y2 <= 2
0 <= (y1 ? y2)<= 2

After introduing slack variables t1,t2,s1,s2, we get

Z                         -y1               = 0
     t1                  - y1 - y2         = 0
                s1      + y1 + y2       = 2
         t2              - y1 + y2        = 0
                    s2 + y1 - y2         = 2

The BFS corresponding to this table

(Z,t1,t2,s1,s2,y1,y2) = (0,0,0,2,2,0,0) corresponds to (Z,x1,x2) =
(0,0,0) in the original LP.

Now a single pivot on y1 in row 3 produces the optimal table

Z            +s1                +y2       = 2
   t1         +s1                            = 2
               s1         +y1  +y2       = 2
       t2    +s1               +2y2      = 2
             -s1  + s2         -2y2      = 0

The BFS corresponding to this table

(Z,t1,t2,s1,s2,y1,y2) = (2,2,2,0,0,2,0) corresponds to (Z,x1,x2) =
( 2,1,1) in the original LP.

Thus, the optimum is reached in one pivot.

Any ideas to generalize to n dimensions would be appreciated.