To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints using mathematical notation.

Let's define:
- $x_1$ as the amount of black beans,
- $x_2$ as the quantity of cornichons.

The objective function is to maximize $2x_1 + 5x_2$, which represents maximizing the value of twice the amount of black beans plus five times the quantity of cornichons.

Given constraints:
1. Tastiness rating constraint: $20x_1 + 13x_2 \geq 102$,
2. Sourness index constraint: $25x_1 + 13x_2 \geq 69$,
3. Mixed constraint: $4x_1 - 4x_2 \geq 0$,
4. Maximum tastiness rating: $20x_1 + 13x_2 \leq 152$,
5. Maximum sourness index: $25x_1 + 13x_2 \leq 240$.

The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'black beans'), ('x2', 'cornichons')],
    'objective_function': 'maximize 2*x1 + 5*x2',
    'constraints': [
        '20*x1 + 13*x2 >= 102',
        '25*x1 + 13*x2 >= 69',
        '4*x1 - 4*x2 >= 0',
        '20*x1 + 13*x2 <= 152',
        '25*x1 + 13*x2 <= 240'
    ]
}
```

Now, let's implement this problem using Gurobi in Python. Note that we need to ensure $x_2$ (cornichons) is an integer variable.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="black_beans")
x2 = m.addVar(vtype=GRB.INTEGER, name="cornichons")

# Set the objective function
m.setObjective(2*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*x1 + 13*x2 >= 102, "tastiness_rating")
m.addConstr(25*x1 + 13*x2 >= 69, "sourness_index")
m.addConstr(4*x1 - 4*x2 >= 0, "mixed_constraint")
m.addConstr(20*x1 + 13*x2 <= 152, "max_tastiness_rating")
m.addConstr(25*x1 + 13*x2 <= 240, "max_sourness_index")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Black beans: {x1.x}")
    print(f"Cornichons: {x2.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```