## Problem Description and Formulation

The problem is an optimization problem with two variables: 'milligrams of vitamin A' and 'milligrams of vitamin B5'. The objective function to minimize is:

\[ 1.66 \times (\text{milligrams of vitamin A})^2 + 2.23 \times (\text{milligrams of vitamin B5})^2 + 5.28 \times \text{milligrams of vitamin B5} \]

The problem has several constraints:

1. **Individual Support Indices**:
   - Milligrams of vitamin A: immune support index = 20, cardiovascular support index = 13.
   - Milligrams of vitamin B5: immune support index = 1, cardiovascular support index = 5.

2. **Combined Support Indices**:
   - Total immune support index \(\geq 32\).
   - Total cardiovascular support index \(\geq 11\).

3. **Bounds on Combined Indices**:
   - Total immune support index \(\leq 54\).
   - Total cardiovascular support index \(\leq 43\).

4. **Quadratic Constraint**:
   - \(5 \times (\text{milligrams of vitamin A})^2 - 4 \times (\text{milligrams of vitamin B5})^2 \geq 0\).

5. **Variable Restrictions**:
   - Milligrams of vitamin A must be a whole number.
   - Milligrams of vitamin B5 can be any real number.

## Gurobi Code Formulation

```python
import gurobipy as gp
from gurobipy import GRB

# Create a new model
model = gp.Model("Vitamin_Optimization")

# Define variables
milligrams_vitamin_A = model.addVar(name="milligrams_vitamin_A", vtype=GRB.INTEGER)
milligrams_vitamin_B5 = model.addVar(name="milligrams_vitamin_B5")

# Objective function
model.setObjective(1.66 * milligrams_vitamin_A**2 + 2.23 * milligrams_vitamin_B5**2 + 5.28 * milligrams_vitamin_B5, GRB.MINIMIZE)

# Constraints
model.addConstr(20 * milligrams_vitamin_A + 1 * milligrams_vitamin_B5 >= 32, name="immune_support_min")
model.addConstr(13 * milligrams_vitamin_A + 5 * milligrams_vitamin_B5 >= 11, name="cardiovascular_support_min")
model.addConstr(20 * milligrams_vitamin_A + 1 * milligrams_vitamin_B5 <= 54, name="immune_support_max")
model.addConstr(13 * milligrams_vitamin_A + 5 * milligrams_vitamin_B5 <= 43, name="cardiovascular_support_max")
model.addConstr(5 * milligrams_vitamin_A**2 - 4 * milligrams_vitamin_B5**2 >= 0, name="quadratic_constraint")

# Solve the model
model.optimize()

# Print the results
if model.status == GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Milligrams of Vitamin A: {milligrams_vitamin_A.varValue}")
    print(f"Milligrams of Vitamin B5: {milligrams_vitamin_B5.varValue}")
    print(f"Objective: {model.objVal}")
else:
    print("The model is infeasible.")
```