Problem #$1($Conceptual Questions$)[20$ points$]$

Please answer "yes" or $"$no$"$ to the following questions. If your answer is no$,$ please state the reason or provide a counter$-$example.

$(1)$ For a given maximization optimization problem, the Simplex algorithm can always make the objective value larger at each iteration.

$(2)$ As long as we select an eligible entering variable and an eligible leaving variable at each iteration, the Simplex algorithm guarantees to terminate in a finite number of steps.

$(3)$ For the worst case, the Simplex algorithm needs to visit every basic feasible solution before it reaches the optimal solution.

$(4)$ At each iteration, the Simplex algorithm moves from one extreme point to another extreme point.

$(5)$ If the primal problem has a feasible solution, then the dual problem must be bounded.

$(6)$ For any given linear program primal problem, there always exists a corresponding dual problem.

$(7)$ If the primal problem is a minimization problem, then the dual problem objective value provides an upper bound for the primal problem.

$(8)$ If the primal problem is a minimization problem with a bounded feasible region, then the dual problem must have an optimal solution.

$(9)$ If the primal problem is infeasible, then the dual problem is unbounded.

$(10)$ If the dual problem is feasible and the objective value is bounded, then the primal problem has an optimal solution.