
Go from any vertex to any other vertex in one Simplex method pivot.
Posted:
Jan 19, 2013 9:46 PM


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.

