I have a linear programming problem in standard form: $min[c^Tx : Ax = b, x ≥ 0]$
Where:
- $A ∈ R^{m×n}$ has full row rank
- $b ∈ R^m$
- $c ∈ R^n$
- $x^* ∈ R^n$ is an optimal nondegenerate vertex with corresponding index sets $B$ and $N$
I'm struggling to prove the statement that if $x^*$ is the unique optimal solution of $(P)$, then the reduced costs of all nonbasic variables are strictly positive?
I've written this statement mathematical terms to try understand it:
$c̅_j = c_j − c^T_B(A_B)^{−1}A_j > 0, \space j ∈ N$