You are given the following linear program (LP) in equational form:

```
min -x₁ -x₂
s.t. 2x₁ + 3x₂ +x₃ = 12 (LP)
x₁ + x₄ = 5
x₁ + 4x₂ +x₅ = 16
xᵢ ≥ 0
```

Consider the point x₂=4,X₄=5,X₁=X₃=X₅=O. Is x feasible for (LP)? Is it a basic feasible solution (BFS)? Is it degenerate?

Consider the point x₁=1,x₂=2,x₃=4,x₄=4,x₅=7. Use the construction of the proof of Theorem 4.2.3 to find a basic feasible solution with objective value no worse than this point.

The

`Minkowski Difference`

`X⊖Y`

of sets`X`

and`Y`

is defined as`{ x - y | x ∈ X, y ∈ Y }`

. Prove that if`X`

and`Y`

are convex then so is`X ⊖ Y`

.Prove that even if

`P`

and`Q`

are convex,`P ∪ Q`

is not necessarily convex.Prove that for any LP, for any point

`z`

, the set of feasible points with the same objective value as`z`

is convex.