the given problem is to show that

if $x_1,...,x_n$ is a nondegenerate basic feasible solution of the primal LP

max $\sum_{j=1}^{n}c_jx_j$

s.t. $\sum_{j=1}^na_{ij}x_j\leq b_i, \forall i\in\{1,...,m\}$

$x_j\geq 0, \forall j\in\{1,...,n\}$

then the problem given as

$\sum_{i=1}^m a_{ij}y_i =c_j$ whenever $x_j>0$

$y_i=0$ whenever $\sum_{j=1}^n a_{ij}x_j < b_i$

has a unique solution.

If $B$ denote the corresponding basis of the given basic solution, letting $y^T= c_B^TB^{-1}$ guarantees the existence of the solution, where $c_B$ denotes the coefficients of the objective functions in the primal corresponding to the given basis.

Now I let $z\neq y$ be a solution of the problem below and tried to induce the contradiction, but could not proceed further. What can I do here? Or is there simpler version rather than this apagogic approach?


I think there are some details missing in your question.

Let the primal be the minimization of $c^Tx$, subject to $Ax = b, x\geq 0$. We will show the following: if the optimal simplex tableau gives us a non-degenerate basic feasible solution with $c_i - z_i \geq 0$ for all variables, then the dual has a unique optimal solution.

Use the complementary slackness conditions.
Note that rank$(A^T_{n \times m})$ = rank$(A_{m \times n}) = m$, a common assumption (because we can just delete rows if redundant). This means the basis matrix $B$ is of dimension $m \times m$.
There are $m$ constraints, and so, $m$ dual variables $y_1, \ldots y_m$.

Consider the basic variable $x_{Bi}$ in this optimal BFS $x$, and an optimal solution $y$ of the dual. As the BFS is non-degenerate, $x_{Bi} > 0$. The complementary slackness conditions give, $x_{Bi} (A^Ty - c)_{Bi} = 0$ which imply, $(A^Ty)_{Bi} = a_{Bi}^Ty = c_{Bi}$. Here, $a_{Bi}^T$ is the row corresponding to the basic variable $x_{Bi}$ in $A^T$. Repeating this for all the $m$ basic variables, we get the system of equations, compactly represented as $(A^Ty)_{B} = c_{B}$. By the definition of a basis, the set of all rows $a_{B1}^T, \ldots, a_{Bm}^T$ are all linearly independent. Thus, there exists only a unique solution $y$ to $(A^Ty)_{B} = c_{B}$.
In fact, multiplying by $(B^{-1})^T$ gives us $y = (B^{-1})^Tc_{B}$ as claimed. Thus, this solution is unique.

We must also ensure that $y$ is feasible for the dual. This comes from the fact that $c_i - z_i = c_i - a_i^Ty \geq 0$ for all $i$. The dual's feasible region is given by $A^Ty \leq c$. So, $y$ is feasible as well. This is what we had to show!

  • $\begingroup$ I wish this came sooner... haha. Thank you anyway! $\endgroup$ – Ian Lee Apr 14 at 15:56
  • $\begingroup$ No problem! Linear programming can be hard! $\endgroup$ – Ameya Apr 14 at 17:14

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