How to find the intersection between a polytope and a hyperplane

[alsm]

Good afternoon!

I am working on a problem, where I at some point have to find the intersection between a polytope and a hyperplan.

Consider the following convex set:

x2>=x1>=x4>=x3
x1+x2+x3+x4=C1

where C1 is a number. In matrixform it can be represented in the following way:

A1*x<=b1,
Aeq*x=beq

where

A1 =

1 -1 0 0
-1 0 0 1
0 0 1 -1

b1 =

0
0
0

Aeq= 1 1 1 1

beq = C1I also have another set:
x1+x4=C2
x1+x2+x3+x4=C1

I matrixform:
Aeq2 * x = beq2

Where

Aeq2=

1 0 0 1
1 1 1 1

beq2=

C2
C1
My problem is that I need to find the intersection between these two sets, i.e. all the points where

x2>=x1>=x4>=x3
x1+x4=C2
x1+x2+x3+x4=C1I used linear algebra a long time ago and have thus forgotten the most of it. Could you please tell me what I have to do or what I have to read to solve my problem. I will have to solve this problem in Matlab once I understand how to solve it.

Thank you very much for your time and consideration!Alex

 

[jvicens]

:

Hello there,

To find the intersection between two sets, you can use a method called linear programming. This involves creating an objective function and constraints that represent the two sets, and then optimizing the objective function to find the points that satisfy both sets.

In your case, your objective function could be something like x1 + x2 + x3 + x4, and your constraints would be the inequalities and equations you listed above. You can then use a linear programming solver in Matlab to find the optimal solution, which will give you the values of x1, x2, x3, and x4 that satisfy both sets.

I recommend reading up on linear programming and how to use it in Matlab to solve your problem. There are many resources available online that can help you with this, such as tutorials, videos, and forums.

Good luck with your research!