```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 LPMax Z = x1 + x2such that0<=x1 <=10<=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 unitsquare under the Simplex method.If you do the variable transformation   y1 = x1 +x2   y2 = x1 ? x2    x1 = (y1 + y2)/2    x2 = (y1 ? y2)/2The LP becomesMax Z  = y1Such that0<= y1+y2 <= 20 <= (y1 ? y2)<= 2After introduing slack variables t1,t2,s1,s2, we getZ                         -y1               = 0     t1                  - y1 - y2         = 0                s1      + y1 + y2       = 2         t2              - y1 + y2        = 0                    s2 + y1 - y2         = 2The 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 tableZ            +s1                +y2       = 2   t1         +s1                            = 2               s1         +y1  +y2       = 2       t2    +s1               +2y2      = 2             -s1  + s2         -2y2      = 0The 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.
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