# Math Help - Optimization problem

1. ## Optimization problem

Hello everybody!

I could really use some help here:

$a \in \mathbb{R}^n, \ e^T = (1,...,1) \in \mathbb{R}^n, \ x \in \mathbb{R}^n$
$b \in \mathbb{R}$

Solve the following optimization problem

$\max \ e^T x$, where $a^T x \le b \mbox{ and } x \ge 0$

I hope, someone knows how to do this.

Best wishes,
Rapha

2. Originally Posted by Rapha
Hello everybody!

I could really use some help here:

$a \in \mathbb{R}^n, \ e^T = (1,...,1) \in \mathbb{R}^n, \ x \in \mathbb{R}^n$
$b \in \mathbb{R}$

Solve the following optimization problem

$\max \ e^T x$, where $a^T x \le b \mbox{ and } x \ge 0$

I hope, someone knows how to do this.

Best wishes,
Rapha
This is a linear program and the maximum occurs at a vertex of the simplex defined by the constraints:

$a^T x \le b \mbox{ and } x \ge 0$

(it may occur at multiple vertices, and so on the line, surface or whatever conecting those vertices, but that is not needed to find the maximum)

CB

3. Hello CaptainBlack,
thanks for your comment.
$\max \ e^T x = \max \ (1,…,1)^T \begin{pmatrix} x_1 ; x_2 ; ... ; x_n \end{pmatrix} = \max \ (x_1 + x_2 + ... +x_n)$

$a^T x = a_1 x_1 + a_2 x_2 + … a_n x_n \le b$
I still do not know how to solve it, I’m sorry, but I don’t understand your hint. :-(

More help would be appreciated.
Rapha

4. I am still working on this problem, but did not find a solution yet. Did anyone else?

Rapha