$[$Optimisation$]$ True or False. Explain reasons for each:

$1)$ If the problem has a nondegenerate optimal solution, then it has a unique optimal solution.

$2)$ If the feasible set is unbounded then the problem is unbounded.

$3)$ If you correctly apply the Simplex Method to a feasible dictionary then the new

dictionary is feasible.

$4)$ If the problem has a unique optimal solution, then this solution is nondegenerate.

$5)$ If any bi is negative then the problem is automatically infeasible.

$6)$ If this problem is bounded then the feasible set is bounded.

$7)$ If this problem has a $($nonempty$)$ bounded feasible set then the problem is bounded.

$8)$ Say m $=4$ and n $=3.$ Then any basic feasible solution to this problem will have at

least $3$ variables that are equal to $0.$

$9)$ If the problem is unbounded then the feasible set is unbounded.

$10)$ If each bi $>=0$ then the problem is feasible.

$11)$ Degeneracy always leads to cycling.

$12)$ Cycling can only occur if some basic solution is degenerate.

$13)$ If the optimal value of the auxiliary problem is $-10$ then the original problem is

infeasible.

$14)$ A problem with $2$ constraints and $3$ decision variables has at most $20$ basic feasible solutions.