## Step 1: Define the symbolic variables and their corresponding natural language objects.
The symbolic variables and their corresponding natural language objects are as follows:
- $x_0$ : cheeseburgers
- $x_1$ : apples
- $x_2$ : cantaloupes
- $x_3$ : oreos

## Step 2: Formulate the objective function using symbolic variables.
The objective function to minimize is: $4x_0^2 + x_0x_1 + 9x_0x_2 + 5x_1x_3 + 4x_2x_3 + 4x_0 + x_1 + 9x_2 + 6x_3$.

## 3: List the constraints in symbolic notation.
Constraints:
1. $4x_0 + 13x_1 + 4x_2 + 12x_3 \leq 257$ (grams of fiber)
2. $15x_0 + 10x_1 + 8x_2 + 8x_3 \leq 252$ (milligrams of iron)
3. $9x_0 + 18x_1 + 10x_2 + 5x_3 \leq 186$ (sourness index)
4. $7x_0 + 11x_1 + 2x_2 + 19x_3 \leq 179$ (grams of fat)
5. $4x_0 + 4x_2 \geq 49$ (grams of fiber from cheeseburgers and cantaloupes)
6. $4x_0 + 13x_1 \geq 30$ (grams of fiber from cheeseburgers and apples)
7. $4x_2 + 12x_3 \geq 62$ (grams of fiber from cantaloupes and oreos)
8. $13^2x_1^2 + 12^2x_3^2 \geq 48$ (grams of fiber from apples squared and oreos squared)
9. $4x_0 + 13x_1 + 4x_2 + 12x_3 \geq 48$ (grams of fiber from all)
10. $8x_2 + 8x_3 \geq 29$ (milligrams of iron from cantaloupes and oreos)
11. $10^2x_1^2 + 8^2x_3^2 \geq 39$ (milligrams of iron from apples squared and oreos squared)
12. $15^2x_0^2 + 8^2x_2^2 \geq 63$ (milligrams of iron from cheeseburgers squared and cantaloupes squared)
13. $15^2x_0^2 + 10^2x_1^2 + 8^2x_3^2 \geq 47$ (milligrams of iron from cheeseburgers squared, apples squared, and oreos squared)
14. $15x_0 + 10x_1 + 8x_2 + 8x_3 \geq 47$ (milligrams of iron from all)
15. $18^2x_1^2 + 10^2x_2^2 \geq 43$ (sourness index from apples squared and cantaloupes squared)
16. $18^2x_1^2 + 5^2x_3^2 \geq 34$ (sourness index from apples squared and oreos squared)
17. $9x_0 + 18x_1 \geq 15$ (sourness index from cheeseburgers and apples)
18. $9x_0 + 18x_1 + 10x_2 + 5x_3 \geq 15$ (sourness index from all)
19. $2x_2 + 19x_3 \geq 29$ (grams of fat from cantaloupes and oreos)
20. $7x_0 + 19x_3 \geq 43$ (grams of fat from cheeseburgers and oreos)
21. $11x_1 + 2x_2 \geq 17$ (grams of fat from apples and cantaloupes)
22. $7x_0 + 2x_2 \geq 42$ (grams of fat from cheeseburgers and cantaloupes)
23. $7x_0 + 11x_1 + 2x_2 \geq 32$ (grams of fat from cheeseburgers, apples, and cantaloupes)
24. $7x_0 + 11x_1 + 2x_2 + 19x_3 \geq 32$ (grams of fat from all)
25. $3x_1 - 7x_2 \geq 0$
26. $x_0^2 - 3x_3^2 \geq 0$
27. $13x_1 + 12x_3 \leq 191$ (grams of fiber from apples and oreos)
28. $13x_1 + 4x_2 \leq 223$ (grams of fiber from apples and cantaloupes)
29. $4^2x_0^2 + 13^2x_1^2 \leq 131$ (grams of fiber from cheeseburgers squared and apples squared)
30. $4^2x_2^2 + 12^2x_3^2 \leq 85$ (grams of fiber from cantaloupes squared and oreos squared)
31. $4x_0 + 4x_2 \leq 226$ (grams of fiber from cheeseburgers and cantaloupes)
32. $10^2x_1^2 + 8^2x_2^2 \leq 75$ (milligrams of iron from apples squared and cantaloupes squared)
33. $7^2x_0^2 + 11^2x_1^2 \leq 131$ (grams of fat from cheeseburgers squared and apples squared)
34. $7x_0 + 11x_1 + 2x_2 \leq 138$ (grams of fat from cheeseburgers, apples, and cantaloupes)

## 4: Provide the symbolic representation of the problem.
```json
{
    'sym_variables': [
        ('x0', 'cheeseburgers'),
        ('x1', 'apples'),
        ('x2', 'cantaloupes'),
        ('x3', 'oreos')
    ],
    'objective_function': '4*x0^2 + x0*x1 + 9*x0*x2 + 5*x1*x3 + 4*x2*x3 + 4*x0 + x1 + 9*x2 + 6*x3',
    'constraints': [
        '4*x0 + 13*x1 + 4*x2 + 12*x3 <= 257',
        '15*x0 + 10*x1 + 8*x2 + 8*x3 <= 252',
        '9*x0 + 18*x1 + 10*x2 + 5*x3 <= 186',
        '7*x0 + 11*x1 + 2*x2 + 19*x3 <= 179',
        '4*x0 + 4*x2 >= 49',
        '4*x0 + 13*x1 >= 30',
        '4*x2 + 12*x3 >= 62',
        '169*x1^2 + 144*x3^2 >= 48',
        '4*x0 + 13*x1 + 4*x2 + 12*x3 >= 48',
        '8*x2 + 8*x3 >= 29',
        '100*x1^2 + 64*x3^2 >= 39',
        '225*x0^2 + 64*x2^2 >= 63',
        '225*x0^2 + 100*x1^2 + 64*x3^2 >= 47',
        '15*x0 + 10*x1 + 8*x2 + 8*x3 >= 47',
        '324*x1^2 + 100*x2^2 >= 43',
        '324*x1^2 + 25*x3^2 >= 34',
        '9*x0 + 18*x1 >= 15',
        '9*x0 + 18*x1 + 10*x2 + 5*x3 >= 15',
        '2*x2 + 19*x3 >= 29',
        '7*x0 + 19*x3 >= 43',
        '11*x1 + 2*x2 >= 17',
        '7*x0 + 2*x2 >= 42',
        '7*x0 + 11*x1 + 2*x2 >= 32',
        '7*x0 + 11*x1 + 2*x2 + 19*x3 >= 32',
        '3*x1 - 7*x2 >= 0',
        'x0^2 - 3*x3^2 >= 0',
        '13*x1 + 12*x3 <= 191',
        '13*x1 + 4*x2 <= 223',
        '16*x0^2 + 169*x1^2 <= 131',
        '16*x2^2 + 144*x3^2 <= 85',
        '4*x0 + 4*x2 <= 226',
        '100*x1^2 + 64*x2^2 <= 75',
        '49*x0^2 + 121*x1^2 <= 131',
        '7*x0 + 11*x1 + 2*x2 <= 138'
    ]
}
```

## Step 5: Implement the optimization problem using Gurobi.
```python
import gurobi

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

    # Define variables
    x0 = model.addVar(name='x0', lb=0)  # cheeseburgers
    x1 = model.addVar(name='x1', lb=0)  # apples
    x2 = model.addVar(name='x2', lb=0)  # cantaloupes
    x3 = model.addVar(name='x3', lb=0)  # oreos

    # Objective function
    model.setObjective(4 * x0**2 + x0 * x1 + 9 * x0 * x2 + 5 * x1 * x3 + 4 * x2 * x3 + 4 * x0 + x1 + 9 * x2 + 6 * x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(4 * x0 + 13 * x1 + 4 * x2 + 12 * x3 <= 257)
    model.addConstr(15 * x0 + 10 * x1 + 8 * x2 + 8 * x3 <= 252)
    model.addConstr(9 * x0 + 18 * x1 + 10 * x2 + 5 * x3 <= 186)
    model.addConstr(7 * x0 + 11 * x1 + 2 * x2 + 19 * x3 <= 179)
    model.addConstr(4 * x0 + 4 * x2 >= 49)
    model.addConstr(4 * x0 + 13 * x1 >= 30)
    model.addConstr(4 * x2 + 12 * x3 >= 62)
    model.addConstr(169 * x1**2 + 144 * x3**2 >= 48)
    model.addConstr(4 * x0 + 13 * x1 + 4 * x2 + 12 * x3 >= 48)
    model.addConstr(8 * x2 + 8 * x3 >= 29)
    model.addConstr(100 * x1**2 + 64 * x3**2 >= 39)
    model.addConstr(225 * x0**2 + 64 * x2**2 >= 63)
    model.addConstr(225 * x0**2 + 100 * x1**2 + 64 * x3**2 >= 47)
    model.addConstr(15 * x0 + 10 * x1 + 8 * x2 + 8 * x3 >= 47)
    model.addConstr(324 * x1**2 + 100 * x2**2 >= 43)
    model.addConstr(324 * x1**2 + 25 * x3**2 >= 34)
    model.addConstr(9 * x0 + 18 * x1 >= 15)
    model.addConstr(9 * x0 + 18 * x1 + 10 * x2 + 5 * x3 >= 15)
    model.addConstr(2 * x2 + 19 * x3 >= 29)
    model.addConstr(7 * x0 + 19 * x3 >= 43)
    model.addConstr(11 * x1 + 2 * x2 >= 17)
    model.addConstr(7 * x0 + 2 * x2 >= 42)
    model.addConstr(7 * x0 + 11 * x1 + 2 * x2 >= 32)
    model.addConstr(7 * x0 + 11 * x1 + 2 * x2 + 19 * x3 >= 32)
    model.addConstr(3 * x1 - 7 * x2 >= 0)
    model.addConstr(x0**2 - 3 * x3**2 >= 0)
    model.addConstr(13 * x1 + 12 * x3 <= 191)
    model.addConstr(13 * x1 + 4 * x2 <= 223)
    model.addConstr(16 * x0**2 + 169 * x1**2 <= 131)
    model.addConstr(16 * x2**2 + 144 * x3**2 <= 85)
    model.addConstr(4 * x0 + 4 * x2 <= 226)
    model.addConstr(100 * x1**2 + 64 * x2**2 <= 75)
    model.addConstr(49 * x0**2 + 121 * x1**2 <= 131)
    model.addConstr(7 * x0 + 11 * x1 + 2 * x2 <= 138)

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"cheeseburgers: {x0.varValue}")
        print(f"apples: {x1.varValue}")
        print(f"cantaloupes: {x2.varValue}")
        print(f"oreos: {x3.varValue}")
    else:
        print("No optimal solution found.")

optimize_problem()
```