To solve the optimization problem described, we will use Gurobi, a powerful linear and mixed-integer programming solver. The objective function to maximize involves variables representing milligrams of vitamins K, B5, and A, with various constraints related to their indices.

Given the complexity of the problem, let's break down the key components:
1. **Objective Function**: Maximize \(3 \times \text{vitamin K} \times \text{vitamin B5} + 3 \times (\text{vitamin B5})^2 + 2 \times \text{vitamin B5} + 3 \times \text{vitamin A}\).
2. **Constraints**:
    - Cognitive, kidney support, cardiovascular support, and immune support indices for each vitamin.
    - Combined minimum and maximum constraints for various combinations of vitamins.

This problem involves quadratic terms (e.g., \((\text{vitamin B5})^2\)), indicating it is not a linear programming problem but rather a quadratic or mixed-integer quadratic programming problem if integer solutions are required. However, since fractional amounts of vitamins are allowed, we treat this as a continuous quadratic programming problem.

Here's the Gurobi code to solve this optimization problem:

```python
from gurobipy import *

# Create a model
m = Model("Vitamin_Optimization")

# Define variables
vitamin_K = m.addVar(name="milligrams_of_vitamin_K", lb=0, ub=GRB.INFINITY)
vitamin_B5 = m.addVar(name="milligrams_of_vitamin_B5", lb=0, ub=GRB.INFINITY)
vitamin_A = m.addVar(name="milligrams_of_vitamin_A", lb=0, ub=GRB.INFINITY)

# Objective function
m.setObjective(3 * vitamin_K * vitamin_B5 + 3 * vitamin_B5**2 + 2 * vitamin_B5 + 3 * vitamin_A, GRB.MAXIMIZE)

# Constraints
# Cognitive performance index constraints
m.addConstr(vitamin_K * 5 + vitamin_B5 * 4 + vitamin_A * 14 >= 32, name="cognitive_combined_min")
m.addConstr(vitamin_K * 5 + vitamin_B5 * 4 + vitamin_A * 14 <= 70, name="cognitive_combined_max")
m.addConstr(vitamin_B5 * 4 + vitamin_A * 14 >= 42, name="cognitive_B5_A_min")
m.addConstr(vitamin_K * 5 + vitamin_A * 14 >= 38, name="cognitive_K_A_min")

# Kidney support index constraints
m.addConstr(vitamin_K**2 * 17 + vitamin_A**2 * 5 >= 30, name="kidney_K_A_squared_min")
m.addConstr(vitamin_B5**2 * 10 + vitamin_A**2 * 5 <= 105, name="kidney_B5_A_squared_max")
m.addConstr(vitamin_K * 17 + vitamin_B5 * 10 + vitamin_A * 5 <= 121, name="kidney_combined_max")

# Cardiovascular support index constraints
m.addConstr(vitamin_K**2 * 14 + vitamin_B5**2 * 17 + vitamin_A**2 * 4 >= 27, name="cardiovascular_combined_min")
m.addConstr(vitamin_K * 14 + vitamin_B5 * 17 <= 68, name="cardiovascular_K_B5_max")
m.addConstr(vitamin_K * 14 + vitamin_A * 4 <= 72, name="cardiovascular_K_A_max")
m.addConstr(vitamin_K * 14 + vitamin_B5 * 17 + vitamin_A * 4 <= 72, name="cardiovascular_combined_max")

# Immune support index constraints
m.addConstr(vitamin_K * 10 + vitamin_B5 * 16 + vitamin_A * 6 >= 59, name="immune_combined_min")
m.addConstr(vitamin_K * 10 + vitamin_B5 * 16 + vitamin_A * 6 <= 102, name="immune_combined_max")
m.addConstr(vitamin_K * 10 + vitamin_A * 6 <= 85, name="immune_K_A_max")
m.addConstr(vitamin_K * 10 + vitamin_B5 * 16 <= 102, name="immune_K_B5_max")

# Optimize
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Vitamin K: {vitamin_K.x}")
    print(f"Milligrams of Vitamin B5: {vitamin_B5.x}")
    print(f"Milligrams of Vitamin A: {vitamin_A.x}")
else:
    print("No optimal solution found")
```