To solve this optimization problem, we'll use the Gurobi Python API. The problem involves minimizing a quadratic objective function subject to several linear and quadratic constraints.

Here's how we can formulate the problem in code:

```python
from gurobipy import *

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

# Define variables
vitamin_A = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_A")
vitamin_B1 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_B1")
vitamin_B2 = m.addVar(lb=-GRB.INFINITY, ub=GRB.INFINITY, vtype=GRB.INTEGER, name="milligrams_of_vitamin_B2")

# Define the objective function
m.setObjective(9.2 * vitamin_A**2 + 3.11 * vitamin_A * vitamin_B1 + 3.0 * vitamin_B2**2 + 4.71 * vitamin_B1 + 2.77 * vitamin_B2, GRB.MINIMIZE)

# Define constraints
m.addConstr(vitamin_A * 10 == 10, name="muscle_growth_index_vitamin_A")
m.addConstr(vitamin_A * 4 == 4, name="cognitive_performance_index_vitamin_A")
m.addConstr(vitamin_B1 * 9 == 9, name="muscle_growth_index_vitamin_B1")
m.addConstr(vitamin_B1 * 12 == 12, name="cognitive_performance_index_vitamin_B1")
m.addConstr(vitamin_B2 * 4 == 4, name="muscle_growth_index_vitamin_B2")
m.addConstr(vitamin_B2 * 8 == 8, name="cognitive_performance_index_vitamin_B2")

# Additional constraints
m.addConstr(vitamin_B1**2 + vitamin_B2**2 >= 46, name="combined_muscle_growth_index_vitamin_B1_and_B2")
m.addConstr(vitamin_A * 10 + vitamin_B2 * 4 >= 27, name="combined_muscle_growth_index_vitamin_A_and_B2")
m.addConstr(vitamin_A * 10 + vitamin_B1 * 9 >= 36, name="combined_muscle_growth_index_vitamin_A_and_B1")
m.addConstr(vitamin_A * 10 + vitamin_B1 * 9 + vitamin_B2 * 4 >= 36, name="total_combined_muscle_growth_index")
m.addConstr(vitamin_A * 4 + vitamin_B2 * 8 >= 14, name="combined_cognitive_performance_index_vitamin_A_and_B2")
m.addConstr(vitamin_B1 * 12 + vitamin_B2 * 8 >= 6, name="combined_cognitive_performance_index_vitamin_B1_and_B2")
m.addConstr(vitamin_A * 4 + vitamin_B1 * 12 + vitamin_B2 * 8 >= 6, name="total_combined_cognitive_performance_index")

# Additional linear constraints
m.addConstr(4 * vitamin_A - 4 * vitamin_B2 >= 0, name="linear_constraint_1")
m.addConstr(vitamin_A * 4 + vitamin_B1 * 12 <= 43, name="cognitive_performance_index_vitamin_A_and_B1_max")
m.addConstr(vitamin_A * 4 + vitamin_B2 * 8 <= 18, name="cognitive_performance_index_vitamin_A_and_B2_max")

# Optimize the model
m.optimize()

```python 
```