Linear program question

I'm trying to prepare for my midterm and I was going over some problems out of my algorithm book but can't seem to figure out 开发者_如何转开发the following problem:

Find necessary and sufficient conditions on the reals a and b under which the linear program

max: x+y
ax + by <=1
x, y =>0

(a) is infeasible. (b) is unbounded. (c) has a finite and unique optimal solution.

here is what I've come up with: for (a), we can add another constraint: ax+by=>5

I'm not sure what to do about b and c.I'm not sure If I'm allowed to change the constraints I'm already given or add new ones.

Any help will be appreciated. Thanks so much advance.

a) I'm not sure if this is possible unless you add a constraint just like you did.
b) if a and b are both less than or equal to zero your problem will be unbounded
c) if a and b are both larger than zero, and they are not equal to each other you will have a unique optimal solution

a. This linear program never infeasible. No matter what value of a and b, there is always a feasible solution to satisfy ax + by <= 1

b. This linear program is unbounded when either a <= 0 or b <= 0.

c. Finite and unique optimal solution exist when a != b and both a > 0 and b > 0

For part (a): it is infeasible when either a=0 and b<0 OR a<0 and b=0