how to solve under-determined system of inequalities with multiply variables

Question

I am trying to find a way to solve system of linear inequalities such as:

c>0
y+c<0
x+c>0
x+y+c<0
w+c>0
w+y+c>0
w+x+c>0
w+x+y+c<0

I have had no luck in finding a fast computational method to solve them. I have tried using wolfram alpha. It works for some sets but not for others. Moreover I also tried solving such systems using matlab's solve function but with no luck. Any help regarding this matter would be very much appreciated.

Answer 1

In general there are infinite solutions to an under determined system. You could however search for the smallest solution of an adapted problem. In that case you could preceed as follows:

You first vectorize your problem

x = [c y x w].';
M = [1 0 0 0
     1 1 0 0
     1 1 1 0
     1 0 0 1
     1 1 0 1
     1 0 1 1
     1 1 1 1];
y = [ 1 -1 1 -1 1 1 1 -1].';

You can set the values of ylike you want. They only need to suffice the conditions of your inequalities (i.e y(1)>0, y(2)<0,...). Now you solve the under determined system Mx=y.

The smallest solution of this system can be found by using the pseudo-inverse of M.

x = M.'(M*M.')^(-1)*y;

If you have chosen yaccordingly to your restrictions, the solution of this problem is also a solution of your problem.

If you want the smallest solution to your problem, just give y only an epsilon room (but then you should calculate it analytical).