To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables and the objective function along with the constraints as per the problem statement.

The problem involves two types of employees: newcomers and full-time employees. We can represent the number of newcomers as \(x_1\) and the number of full-time employees as \(x_2\).

Given:
- Newcomers earn $400 a week.
- Full-time employees earn $700 a week.
- The company needs at least 100 gardeners in total.
- At least 40 must be full-time employees.
- The number of full-time employees should be at least half the number of newcomers.

Let's formulate this into a symbolic representation:
```json
{
    'sym_variables': [('x1', 'number of newcomers'), ('x2', 'number of full-time employees')],
    'objective_function': '400*x1 + 700*x2',
    'constraints': [
        'x1 + x2 >= 100',  # Total gardeners should be at least 100
        'x2 >= 40',  # At least 40 full-time employees
        'x2 >= 0.5*x1'  # Full-time employees should be at least half the number of newcomers
    ]
}
```

Now, let's translate this into Gurobi code in Python to find the optimal solution:

```python
from gurobipy import *

# Create a new model
m = Model("Gardening_Company")

# Define variables
x1 = m.addVar(name='newcomers', vtype=GRB.CONTINUOUS, lb=0)  # Number of newcomers
x2 = m.addVar(name='full_time_employees', vtype=GRB.CONTINUOUS, lb=0)  # Number of full-time employees

# Set the objective function: Minimize wage bill
m.setObjective(400*x1 + 700*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1 + x2 >= 100, name='total_gardeners')  # Total gardeners should be at least 100
m.addConstr(x2 >= 40, name='min_full_time_employees')  # At least 40 full-time employees
m.addConstr(x2 >= 0.5*x1, name='full_time_newcomer_ratio')  # Full-time employees should be at least half the number of newcomers

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print('Optimal solution found:')
    print(f'Number of newcomers: {x1.x}')
    print(f'Number of full-time employees: {x2.x}')
    print(f'Total wage bill: ${400*x1.x + 700*x2.x:.2f}')
else:
    print('No optimal solution found')
```