Linear Optimization Problem with a constraint on the cost functin Is there a known algorithm ( similar to simplex algorithm) that solves the following problem: 
Maximize $c^Tx$ subject to the constraints $Ax\leq b$ and $c^Tx\leq \alpha$. It would be nice if you can provide a link to an article or software implementation of such problems. 
Thanks
 A: Well of course, all you do is add $c^T$ as a row to your $A$ matrix, and $\alpha$ to your $b$ vector. That is,
$$\bar{A}\triangleq \begin{bmatrix} A \\ c^T \end{bmatrix} \quad \bar{b}\triangleq \begin{bmatrix} b \\ \alpha \end{bmatrix}$$
And then you solve using any preferred algorithm:
$$\begin{array}{ll}
\text{maximize} & c^T x \\
\text{subject to} & \bar{A} x \leq \bar{b}
\end{array}$$
Here's how this works: let $\alpha_{\max}$ be the solution to the original problem (without the extra $c^Tx\leq\alpha$ constraint), and let $\alpha_{\min}$ be the solution to the minimization $$\begin{array}{ll}
\text{minimize} & c^T x \\
\text{subject to} & A x \leq b
\end{array}$$
(Note: it's possible one of these extremes will be infinite.)
Now select any value $\alpha\in[\alpha_{\min},\alpha_{\max}]$ and add your extra constraint $c^Tx\leq \alpha$ as I've shown above. When you solve this modified model, the optimal point $x^*$ will achieve exactly $c^Tx^*=\alpha$.
Again, I'm sorry for misreading the problem earlier! I hope this clarifies matters.
