How to solve a system of linear equations with strict inequalities and equalities?

Hi all. I am looking to solve systems of linear inequalites that is similiar to the below:

F1x = 10x1 + 2x2 + 10x3

F2x = 15x1 + 20x2 + 8x3

F3x = 6x1 + 4x2 + 12x3

System:

F1x - F2x > 0

F1x - F3x > 0

x1, x2, x3 >=0

x1, x2, x3 <= 1

x1 + x2 + x3 = 1

I do not need to find all solutions, I simply need to know whether there is a solution or not.

Thanks!

Answer 1

[David Goodmanson]

Hi Terri,

Looks like it is not possible. You have

[F1;F2;F3] = [10 2 10
              15 20 8
               6 4 12] * [x1;x2;x3]

Now if F1-F2 > 0         ident2
   and F1-F3 > 0         ident3
   then 2*F1-F2-F3 > 0   ident4      necessary but not sufficient

but that last quantity is [2 -1 -1]*[F1;F2;F3]
= [2 -1 -1]*[10 2 10
             15 20 8
             6 4 12] * [x1;x2;x3];

= [-1 -20 0] * [x1;x2;x3]

and that quantity can't be greater than zero. If you had allowed equality with the F's

F1-F2 >= 0 and F1-F3 >= 0

then x1 = x2 = 0, x3 = 1 might have been possible but it gives

F1 = 10   F2 = 8   F3 = 12

which satisfies the equality-allowed version of ident4 but not the equality-allowed version of ident3. So it goes.