Show if true or a counterexample if false
a) If the feasible polytope described by the solution space of a linear programming problem is unbounded then there is no optimal solution
b) If there are two basic optimal feasible solutions then there is an infinity of
optimal feasible solutions.
c) If there is a feasible solution to a linear programming problem then there is
a feasible basic solution to the problem.
a) I think is true because if I try to imagine an unbounded polytope there will be no vertices in it.
b) Again, if I imagain it graphicaly I can see that two vertices make a infinite line of optimal feasible solutions.
Any suggestions or hints for the proofs would be great!
Best Answer
Let me answer with two pictures:
a)
Think of an infinitely large pyramid (or cone). It is unbounded, but has a vertex, and we can optimize in the direction of that vertex and so this vertex can be a feasible optimal solution.
c)
Take this infinitely long "bar". If you optimize "upwards", then all points of the upper edge are optimal, but none of these optimal points is a vertex (aka. basic).