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

- $x_1$ as the number of plates of Vietnamese food.
- $x_2$ as the number of plates of Korean food.

The objective function is to minimize the total cost, which can be represented as:
\[12.5x_1 + 16.5x_2\]

The constraints are:

1. The minimum protein requirement: \[15x_1 + 10x_2 \geq 250\]
2. The minimum carb requirement: \[20x_1 + 14x_2 \geq 45\]
3. Non-negativity constraints for $x_1$ and $x_2$: 
   - $x_1 \geq 0$
   - $x_2 \geq 0$

This symbolic representation can be summarized as:

```json
{
    'sym_variables': [('x1', 'number of plates of Vietnamese food'), ('x2', 'number of plates of Korean food')],
    'objective_function': '12.5*x1 + 16.5*x2',
    'constraints': ['15*x1 + 10*x2 >= 250', '20*x1 + 14*x2 >= 45', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

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

# Create variables
x1 = m.addVar(lb=0, name="Vietnamese_Food")
x2 = m.addVar(lb=0, name="Korean_Food")

# Set the objective function
m.setObjective(12.5*x1 + 16.5*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(15*x1 + 10*x2 >= 250, "Protein_Requirement")
m.addConstr(20*x1 + 14*x2 >= 45, "Carb_Requirement")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Minimum cost: $", round(m.objVal, 2))
    print("Number of plates of Vietnamese food:", round(x1.x))
    print("Number of plates of Korean food:", round(x2.x))
else:
    print("The model is infeasible")
```

```python
from gurobipy import *

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

# Create variables
x1 = m.addVar(lb=0, name="Vietnamese_Food")
x2 = m.addVar(lb=0, name="Korean_Food")

# Set the objective function
m.setObjective(12.5*x1 + 16.5*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(15*x1 + 10*x2 >= 250, "Protein_Requirement")
m.addConstr(20*x1 + 14*x2 >= 45, "Carb_Requirement")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Minimum cost: $", round(m.objVal, 2))
    print("Number of plates of Vietnamese food:", round(x1.x))
    print("Number of plates of Korean food:", round(x2.x))
else:
    print("The model is infeasible")
```