To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables for each of the objects mentioned (milligrams of iron, grams of fiber, milligrams of vitamin A), expressing the objective function using these variables, and listing all constraints in terms of these variables.

Let's define:
- \(x_1\) as the amount of milligrams of iron,
- \(x_2\) as the amount of grams of fiber,
- \(x_3\) as the amount of milligrams of vitamin A.

The objective function to maximize is given by:
\[1 \cdot x_1^2 + 3 \cdot x_1 \cdot x_2 + 4 \cdot x_1 \cdot x_3 + 9 \cdot x_2 \cdot x_3 + 7 \cdot x_3^2 + 3 \cdot x_2\]

The constraints are:
1. Cognitive performance index of milligrams of iron is 27: \(x_1 = 27\)
2. Grams of fiber each have a cognitive performance index of 14: \(x_2 = 14\)
3. Milligrams of vitamin A each have a cognitive performance index of 1: \(x_3 = 1\)
4. Total combined cognitive performance index from milligrams of iron and milligrams of vitamin A is at least 58: \(27 + x_3 \geq 58\)
5. Total combined cognitive performance index from milligrams of iron, grams of fiber, and milligrams of vitamin A is at least 62: \(27 + 14 + x_3 \geq 62\)
6. Total combined cognitive performance index from milligrams of iron squared and milligrams of vitamin A squared is at most 177: \(x_1^2 + x_3^2 \leq 177\)
7. Total combined cognitive performance index from milligrams of iron, grams of fiber, and milligrams of vitamin A is no more than 177: This constraint seems to be inconsistent with the previous ones as it would limit the total to a value less than some of the individual components' minimum requirements based on the problem description.

Given these definitions and constraints, we can represent the optimization problem symbolically as follows:
```json
{
    'sym_variables': [('x1', 'milligrams of iron'), ('x2', 'grams of fiber'), ('x3', 'milligrams of vitamin A')],
    'objective_function': '1*x1**2 + 3*x1*x2 + 4*x1*x3 + 9*x2*x3 + 7*x3**2 + 3*x2',
    'constraints': [
        'x1 = 27',
        'x2 >= 0',  # Since x2 must be a whole number, but its exact value isn't specified, we use an inequality
        'x3 >= 0',  # Similar reasoning for x3
        '27 + x3 >= 58',
        '27 + 14 + x3 >= 62',
        'x1**2 + x3**2 <= 177',
        'x2 == int(x2)'  # To ensure x2 is a whole number, though this will be handled in the Gurobi model
    ]
}
```

Now, let's proceed to implement this optimization problem using Gurobi. Note that since some constraints and the objective function involve quadratic terms, we'll use the `gurobipy` module with its quadratic programming capabilities.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, name="milligrams_of_iron")
x2 = m.addVar(vtype=GRB.INTEGER, lb=0, name="grams_of_fiber")  # Ensure x2 is an integer
x3 = m.addVar(lb=0, name="milligrams_of_vitamin_A")

# Objective function: Maximize
m.setObjective(1*x1**2 + 3*x1*x2 + 4*x1*x3 + 9*x2*x3 + 7*x3**2 + 3*x2, GRB.MAXIMIZE)

# Constraints
m.addConstr(x1 == 27, "iron_constraint")
m.addConstr(27 + x3 >= 58, "iron_vitaminA_min")
m.addConstr(41 + x3 >= 62, "all_min")  # Simplified from 27 + 14 + x3 >= 62
m.addConstr(x1**2 + x3**2 <= 177, "squared_max")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of iron: {x1.x}")
    print(f"Grams of fiber: {x2.x}")
    print(f"Milligrams of vitamin A: {x3.x}")
else:
    print("No optimal solution found.")
```