To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's define two variables: 

- $x_1$ representing the number of waiters
- $x_2$ representing the number of managers

The objective function is to minimize the total wage bill, which can be represented algebraically as $1200x_1 + 2000x_2$.

Now, let's list the constraints based on the problem description:
1. The restaurant requires a minimum of 50 workers: $x_1 + x_2 \geq 50$
2. At least 15 must be managers: $x_2 \geq 15$
3. The number of managers should be at least a third of the number of waiters: $x_2 \geq \frac{1}{3}x_1$
4. The weekly wage bill should be below $500,000: $1200x_1 + 2000x_2 \leq 500,000$

Given this information, we can represent the problem symbolically as follows:
```json
{
    'sym_variables': [('x1', 'number of waiters'), ('x2', 'number of managers')], 
    'objective_function': '1200*x1 + 2000*x2', 
    'constraints': ['x1 + x2 >= 50', 'x2 >= 15', 'x2 >= (1/3)*x1', '1200*x1 + 2000*x2 <= 500000']
}
```

Now, let's implement this optimization problem using Gurobi in Python:
```python
from gurobipy import *

# Create a new model
model = Model("FastFoodRestaurant")

# Define the variables
x1 = model.addVar(vtype=GRB.INTEGER, name="waiters")
x2 = model.addVar(vtype=GRB.INTEGER, name="managers")

# Set the objective function to minimize
model.setObjective(1200*x1 + 2000*x2, GRB.MINIMIZE)

# Add constraints
model.addConstr(x1 + x2 >= 50, "total_workers")
model.addConstr(x2 >= 15, "min_managers")
model.addConstr(x2 >= (1/3)*x1, "manager_waiter_ratio")
model.addConstr(1200*x1 + 2000*x2 <= 500000, "wage_bill")

# Optimize the model
model.optimize()

# Print the results
if model.status == GRB.OPTIMAL:
    print(f"Optimal solution: {model.objVal} with {x1.varName} = {x1.x} and {x2.varName} = {x2.x}")
else:
    print("No optimal solution found")
```