To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints in terms of these variables.

Let's define:
- $x_1$ as the number of hours the university cafe is run.
- $x_2$ as the number of hours the downtown cafe is run.

The objective is to minimize the total cost, which can be represented by the function:
\[400x_1 + 700x_2\]

The constraints are based on the production requirements:
- At least 900 cappuccinos: $30x_1 + 40x_2 \geq 900$
- At least 700 lattes: $40x_1 + 70x_2 \geq 700$
- At least 1400 regular coffees: $60x_1 + 110x_2 \geq 1400$

Additionally, since the cafes cannot be run for a negative number of hours, we have:
- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'hours university cafe is run'), ('x2', 'hours downtown cafe is run')],
    'objective_function': '400*x1 + 700*x2',
    'constraints': [
        '30*x1 + 40*x2 >= 900',
        '40*x1 + 70*x2 >= 700',
        '60*x1 + 110*x2 >= 1400',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

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

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

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_university_cafe")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_downtown_cafe")

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

# Add constraints
m.addConstr(30*x1 + 40*x2 >= 900, "cappuccino_constraint")
m.addConstr(40*x1 + 70*x2 >= 700, "latte_constraint")
m.addConstr(60*x1 + 110*x2 >= 1400, "regular_coffee_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"University cafe should be run for {x1.x} hours")
    print(f"Downtown cafe should be run for {x2.x} hours")
    print(f"Total cost: {m.objVal}")
else:
    print("No optimal solution found")

```