# Ways to solve a linear optimization problem

What are the methods/techniques that one can use to solve linear(or integer) optimization problems ?

Googling around I stumbled upon the Simplex Method. But is that the only method used?

https://www.youtube.com/watch?v=2ACJ9ewUC6U <--- Is this an application of the Simplex Method ?

Let's take Excel Solver for instance. What method is this program using to tackle linear optimization problems?

Which are the techniques that we use today ?

• If you have only two variables you can solve an LP graphically. But in the video on step has been omitted. The objective function can be used in the graph to identify the optimal solution. If you want to know how it work leave a comment. – callculus Feb 28 '17 at 7:33
• Yes please, I would like to know how that works. – HashWizard Feb 28 '17 at 19:13
• I have posted an explanation. If something is not clear feel free to ask. I hope the uploads of the graphs don´t disappear after a while. It was the first time that I had problems to upload them. – callculus Mar 1 '17 at 4:26

Simplex is still a competitive method for many types of problems. That is the method that the Excel Solver uses.

I didn't watch the whole video but it doesn't look like they're using Simplex method - it looks like they are just graphing the constraints and identifying extreme points. There's a theorem that says if an optimal solution exists, then one of the extreme points is an optimal solution, so that's what they would be using there.

There are other methods for solving LPs. For very large problems, interior point methods are the favored method. These methods travel along the interior of the feasible region rather than along the extreme points. See for example this .pdf from MIT.

In the video the graph is almost like the graph below:

Additionally I have drawn the objective function (blue line). To do this you have to solve the objective function, at level $C=0$, for $y$.

$C=0.8x+0.5y$

$0=0.8x+0.5y$

$-0.8x=0.5y$

$-1.6x=y$

You need two points to draw it. One point is obviously $(0/0)$ and another point is for instance $(2/-3.2)$

Now you push the line right upwards parallel until the line touches the feasible region the first time.

The second picture shows the process.

The line touches the feasible region first at $(2.4/1.2)$. If you cannot identify the exact value from the graph you calculate the intersection of the two lines. Here they are the two constraints. But in many cases the intersection can be identified directly from the graph.