$($e$)$ Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP relaxation's optimal objective function value?

Additional insteger constraints allow the other constraints of the LP relaxation to be ignored.

The optimal solution to the ILP is always found by rounding up the optimal solution to the LP relaxation.

None of the feasible solutions to the LP relaxation are feasible solutions to the ILP.

The optimal solution to the ILP is always found by rounding down the optimal solution to the LP relaxation.

Additional integer constraints restrict the feasible region further.

When would the optimal objective function values be equal?

They would be equal if the optimal solution to the ILP is a feasible solution to the LP relaxation.

They would be equal if the optimal solution to the ILP is the same as the optimal solution to the LP relaxation, except rounded up$.$

They would be equal if the optimal solution to the ILP is the same as the optimal solution to the LP relaxation, expst rounded down.

They would be equal if the LP relaxation only uses integer values in the objective function and constraints.

They would be equal if the LP relaxation has an optimal solution with integer values.