# Homework Help: Prove nonempty, unbounded feasible region has no optimal solution

1. Feb 9, 2012

### csc2iffy

1. The problem statement, all variables and given/known data

Let C be the set of all points (x,y) in the plane satisfying x≥0, y≥0, -x-2y≥-8.

a. Show that C is nonempty and unbounded.
b. Prove that the LP problem: Max M=2x+3y subject to the constraint that (x,y) lie in C has no feasible, optimal solution.
c. Show that the LP problem: Max M=-3x-6y subject to the constraint that (x,y) lie in C does have a feasible, optimal solution.

2. Relevant equations

3. The attempt at a solution

a. I graphed the constraints and showed that the feasible region is the entire first quadrant, and therefore C is nonempty and unbounded (provided attachment of my work - is this enough?)

b. I could "show" this but I have no idea how to "prove" it. Does it involve the simplex method?

c. Same question as above

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2. Feb 9, 2012

### Ray Vickson

There is something wrong with the question: what it is asking you is false if you have copied down the constraints correctly. The constraints are x ≥ 0, y ≥ 0 and -x-2y ≥ -8, which is the *same as* x + 2y ≤ 8. Here the feasible region is nonempty and bounded, and so the problems max 2x + 3y and max -3x - 6y (which is the same as min 3x + 6y) both have feasible, optimal solutions.

RGV

3. Feb 9, 2012

### csc2iffy

Sorry here is the problem, corrected:
Let C be the set of all points (x,y) in the plane satisfying x≥0, y≥0, -x-2y≤-8.

a. Show that C is nonempty and unbounded.
b. Prove that the LP problem: Max M=2x+3y subject to the constraint that (x,y) lie in C has no feasible, optimal solution.
c. Show that the LP problem: Max M=-3x-6y subject to the constraint that (x,y) lie in C does have a feasible, optimal solution.

4. Feb 9, 2012

5. Feb 9, 2012

### Ray Vickson

I have already helped, but you have ignored my assistance.

RGV

6. Feb 9, 2012

### csc2iffy

No, I had the problem written down wrong. You helped me with the wrong problem