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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 = C1

I 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=C1

I 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