simplex algorithm - importance of basic solution?

Question

Importance of basic solution in simplex algorithm?

Answer 1

If all variables (structural and logical) are non-negative (i.e. x>=0 and slacks s>=0) then all non-basic variables are equal to zero. As they are fixed to zero we only have to solve for the m basic variables.

Essentially we have to solve

A x = b

Unfortunately this is a non-square system of equations (after adding slacks we always have more columns than rows). In LPs we can form a basic solution and partition this into

B x_B + N x_N = b

After setting x_N = 0 we have just a square system of linear equations with solution:

x_B = inv(B) b

There is a fundamental theorem that says we can restrict the search to only basic solutions i.e. solutions that can be partitioned in basic and non-basic variables

x = [ x_B ]
    [ x_N ]

with x_B >= 0 and x_N = 0.

For more info open a book about Linear Programming; a very good one is Vanderbei.