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Being $A$ a symmetric matrix in

min $c^Tx$

subject to $Ax \geqslant c$

$x \geqslant 0$

I need to show that, if $x^*$ satisfies $Ax^*=c$ and $x^*>0$, so $x^*$ is an optimal solution.

I've tried to show it by the Complementary Slackness Theorem and got stuck in the second equation $(c_j-p^TA_j)x_j=0, \forall j$, since I have no information about $p$, the solution of the dual problem. So, can I go further in this way or there is a better option to show it?

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    Do we have more information on $A$. It seems like you were going to write something in the first line. – 2017-02-11
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    $A$ symmetric is the only thing the exercise says – 2017-02-11
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    what do you know about LP duality theories? What can you say about the primal and dual optimal values? – 2017-02-12

1 Answers 1

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Take the dual of this problem, you get:

max $c^Ty$

$Ay\leq c$

$y\geq 0$

So, $x^*$ is feasible for this problem. However by standard duality theory, a feasible solution to both the primal and the dual is necessarily optimal.

The $x^*>0$ condition isn't really important here, $x^*$ can only be $0$ when $c=0$ which is a degenerate case of this problem.