## Step 1: Define the symbolic representation of the variables
The variables are ['cantaloupes', 'chicken drumsticks', 'apple pies', 'bowls of pasta'] which can be represented symbolically as ['x1', 'x2', 'x3', 'x4'].

## Step 2: Create a dictionary mapping symbolic variables to natural language objects
The symbolic variables and their corresponding natural language objects are:
- x1: cantaloupes
- x2: chicken drumsticks
- x3: apple pies
- x4: bowls of pasta

## Step 3: Formulate the objective function using symbolic variables
The objective function to minimize is: $3.28x_1^2 + 6.66x_1x_3 + 1.42x_1x_4 + 6.84x_2^2 + 9.64x_2x_4 + 3.22x_3^2 + 4.23x_3x_4 + 2.46x_2$.

## 4: Define the constraints
Constraints:
1. $6x_1 + 15x_2 + 18x_3 + 21x_4 \leq 263$
2. $x_1^2 + x_4^2 \geq 44$
3. $x_3^2 + x_4^2 \geq 64$
4. $15x_2 + 18x_3 + 21x_4 \geq 35$
5. $6x_1 + 15x_2 + 21x_4 \geq 35$
6. $15x_2 + 18x_3 + 21x_4 \geq 61$
7. $6x_1 + 15x_2 + 21x_4 \geq 61$
8. $6x_1 + 15x_2 + 18x_3 + 21x_4 \geq 61$
9. $-3x_3^2 + 2x_4^2 \geq 0$
10. $6x_1 + 15x_2 \leq 175$
11. $6^2x_1^2 + 15^2x_2^2 + 21^2x_4^2 \leq 215^2$

## 5: Consider variable bounds and types
- x1, x3 are integers
- x2, x4 are non-negative integers (or more specifically, x2 is nonfractional and x4 is nonfractional, implying they are integers)

## 6: Write down the problem in JSON format
```json
{
    'sym_variables': [
        ('x1', 'cantaloupes'), 
        ('x2', 'chicken drumsticks'), 
        ('x3', 'apple pies'), 
        ('x4', 'bowls of pasta')
    ], 
    'objective_function': '3.28*x1^2 + 6.66*x1*x3 + 1.42*x1*x4 + 6.84*x2^2 + 9.64*x2*x4 + 3.22*x3^2 + 4.23*x3*x4 + 2.46*x2', 
    'constraints': [
        '6*x1 + 15*x2 + 18*x3 + 21*x4 <= 263',
        'x1^2 + x4^2 >= 44',
        'x3^2 + x4^2 >= 64',
        '15*x2 + 18*x3 + 21*x4 >= 35',
        '6*x1 + 15*x2 + 21*x4 >= 35',
        '15*x2 + 18*x3 + 21*x4 >= 61',
        '6*x1 + 15*x2 + 21*x4 >= 61',
        '6*x1 + 15*x2 + 18*x3 + 21*x4 >= 61',
        '-3*x3^2 + 2*x4^2 >= 0',
        '6*x1 + 15*x2 <= 175',
        '36*x1^2 + 225*x2^2 + 441*x4^2 <= 46225'
    ]
}
```

## 7: Implement the problem using Gurobi
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name="cantaloupes", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="chicken_drumsticks", vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name="apple_pies", vtype=gurobi.GRB.INTEGER)
    x4 = model.addVar(name="bowls_of_pasta", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(3.28*x1**2 + 6.66*x1*x3 + 1.42*x1*x4 + 6.84*x2**2 + 9.64*x2*x4 + 3.22*x3**2 + 4.23*x3*x4 + 2.46*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(6*x1 + 15*x2 + 18*x3 + 21*x4 <= 263)
    model.addConstr(x1**2 + x4**2 >= 44)
    model.addConstr(x3**2 + x4**2 >= 64)
    model.addConstr(15*x2 + 18*x3 + 21*x4 >= 35)
    model.addConstr(6*x1 + 15*x2 + 21*x4 >= 35)
    model.addConstr(15*x2 + 18*x3 + 21*x4 >= 61)
    model.addConstr(6*x1 + 15*x2 + 21*x4 >= 61)
    model.addConstr(6*x1 + 15*x2 + 18*x3 + 21*x4 >= 61)
    model.addConstr(-3*x3**2 + 2*x4**2 >= 0)
    model.addConstr(6*x1 + 15*x2 <= 175)
    model.addConstr(36*x1**2 + 225*x2**2 + 441*x4**2 <= 46225)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("Cantaloupes: ", x1.varValue)
        print("Chicken drumsticks: ", x2.varValue)
        print("Apple pies: ", x3.varValue)
        print("Bowls of pasta: ", x4.varValue)
    else:
        print("No solution found")

solve_optimization_problem()
```