Look carefully at the problem you have posed. Is there some direction we can move infinitely far out, and still obey those constraints?
Your objective function is of the form:
A = [-1 -1 -1; 2 0 -1;0 -2 1];
b = [-6;-2;0];
lb = [0 0 0];
You have linear inequality constraints of the form A*x <= b.
You wish to maximize an objective of the form dot(f,x), where f is the vector
f = [2 1 2];
Does a feasible point exist? We can actually find one by simply changing all of your constraints to equalities. Does a trivially feasible solution exist?
xfeas = (A\b)'
Happily, it even obeys your bound constraints, since x1,x2,x3 are all positive. So xfeas is indeed a feasible solution to the problem.
format rat
f = [2 -1 2];
dot(f,xfeas)
The probem is, there exists a direction we can move which will increase the objective as large as we wish, yet still remains feasible always. That is the meaning of unbounded.
dx = [1 1 2];
syms t
dot(f,xfeas + dx*t)
Of course, when t == 0, we get the original point. And for any value of t>0, 5*t+27/4 is always greater then 27/4.
A*(xfeas + dx*t).' - b
So as t grows, the constraints are still maintained. And for all positive t, the bound constraints are also maintained.
xfeas + dx*t
Linprog is correct. Your problem is unbounded. Could I have spent some time writing out the dual to this problem, and explaining how that would have helped in this analysis? Probably. But that would have required I explain what the dual is and why it would help.
I spent some time writing out a lot of explanations abut linear programming, and unbounded problems, etc., here:
