Calculate the family of optimization models


Assignment:

Consider the family of optimization models:

         max x1 + x2

         s.t.  x1 + x2 ≤ t

 (Pt)   x21 + x22 ≤ 1/1 + t2

         x1 ≥ 0, x2 ≥ 0,

where t is a parameter (it is not a variable)

1. Solve (Pt) when t = 1.

2. For what values of t is the solution (1,0) feasible for (Pt)?

3. For what values of t is (Pt) infeasible?

4. For what values of t is the feasible region of (Pt) convex? (Technically, an empty set is convex)

5. For what values of t does (Pt) have a single optimal solution.

6. For what values of t is the solution (t, 0) optimal for (Pt)

7. For what values of t does (Pt) have an infinite number of solutions, but (t, 0) is not one of them. (It might be useful to observe that the equation at4  + bt2 + c= 0 can be easily solved if you know how to solve ax2 + bx + c = 0).

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