To solve this optimization problem, we first need to define the variables and the objective function, and then add all the constraints as described.

## Step 1: Define the Variables
Let's denote 'milligrams of vitamin D' as $x_0$, 'milligrams of vitamin B9' as $x_1$, and 'milligrams of zinc' as $x_2$.

## 2: Define the Objective Function
The objective function to minimize is $3.53x_0 + 5.16x_1 + 2.52x_2$.

## 3: Define the Constraints
We have the following constraints:
1. $5x_0 \leq 198$
2. $x_0 \leq 135$
3. $2x_0 \leq 174$
4. $6x_1 \leq 198$
5. $4x_1 \leq 135$
6. $7x_1 \leq 174$
7. $11x_2 \leq 198$
8. $2x_2 \leq 135$
9. $11x_2 \leq 174$
10. $6x_1 + 11x_2 \geq 36$
11. $5x_0 + 6x_1 + 11x_2 \geq 36$
12. $x_0 + 4x_1 \geq 44$
13. $x_0 + 4x_1 + 2x_2 \geq 44$
14. $2x_0 + 7x_1 \geq 20$
15. $2x_0 + 11x_2 \geq 33$
16. $2x_0 + 7x_1 + 11x_2 \geq 33$
17. $7x_0 - 10x_2 \geq 0$
18. $9x_1 - 3x_2 \geq 0$
19. $-5x_0 + 8x_1 \geq 0$
20. $5x_0 + 6x_1 \leq 150$
21. $2x_0 + 11x_2 \leq 164$
22. $7x_1 + 11x_2 \leq 173$

## 4: Implement in Gurobi
We will use Gurobi to solve this linear programming problem.

```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x0 = model.addVar(name="milligrams_of_vitamin_D", vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name="milligrams_of_vitamin_B9")
    x2 = model.addVar(name="milligrams_of_zinc")

    # Define the objective function
    model.setObjective(3.53 * x0 + 5.16 * x1 + 2.52 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(5 * x0 <= 198)
    model.addConstr(x0 <= 135)
    model.addConstr(2 * x0 <= 174)
    model.addConstr(6 * x1 <= 198)
    model.addConstr(4 * x1 <= 135)
    model.addConstr(7 * x1 <= 174)
    model.addConstr(11 * x2 <= 198)
    model.addConstr(2 * x2 <= 135)
    model.addConstr(11 * x2 <= 174)
    model.addConstr(6 * x1 + 11 * x2 >= 36)
    model.addConstr(5 * x0 + 6 * x1 + 11 * x2 >= 36)
    model.addConstr(x0 + 4 * x1 >= 44)
    model.addConstr(x0 + 4 * x1 + 2 * x2 >= 44)
    model.addConstr(2 * x0 + 7 * x1 >= 20)
    model.addConstr(2 * x0 + 11 * x2 >= 33)
    model.addConstr(2 * x0 + 7 * x1 + 11 * x2 >= 33)
    model.addConstr(7 * x0 - 10 * x2 >= 0)
    model.addConstr(9 * x1 - 3 * x2 >= 0)
    model.addConstr(-5 * x0 + 8 * x1 >= 0)
    model.addConstr(5 * x0 + 6 * x1 <= 150)
    model.addConstr(2 * x0 + 11 * x2 <= 164)
    model.addConstr(7 * x1 + 11 * x2 <= 173)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin D: {x0.varValue}")
        print(f"Milligrams of vitamin B9: {x1.varValue}")
        print(f"Milligrams of zinc: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```