## Step 1: Define the variables and the objective function
The variables are ['milligrams of vitamin B7', 'milligrams of vitamin B12', 'milligrams of zinc', 'milligrams of vitamin B2']. Let's denote them as $x_0, x_1, x_2, x_3$ respectively.

The objective function to maximize is:
\[ 7.67x_0^2 + 1.72x_0x_1 + 8.57x_0x_3 + 8.81x_1^2 + 2.9x_1x_2 + 7.6x_2^2 + 1.36x_2x_3 + 5.69x_0 + 5.34x_2 \]

## Step 2: Define the constraints
The constraints given are:
- $10x_0 + 11x_1 + 13x_2 + 12x_3 \leq 83$ (energy stability index)
- $9x_0 + 5x_1 + 10x_2 + 11x_3 \leq 312$ (cognitive performance index)
- $11x_1 + 13x_2 \geq 14$ (energy stability index from $x_1, x_2$)
- $11x_1 + 12x_3 \geq 10$ (energy stability index from $x_1, x_3$)
- $10x_0^2 + 13x_2^2 \geq 8$ (energy stability index from $x_0^2, x_2^2$)
- $13x_2 + 12x_3 \geq 8$ (energy stability index from $x_2, x_3$)
- $10x_0 + 11x_1 \geq 12$ (energy stability index from $x_0, x_1$)
- $9x_0 + 5x_1 + 10x_2 \geq 67$ (cognitive performance index from $x_0, x_1, x_2$)
- $9x_0 + 5x_1 + 11x_3 \geq 67$ (cognitive performance index from $x_0, x_1, x_3$)
- $9x_0^2 + 5x_1^2 + 10x_2^2 \geq 49$ (cognitive performance index from $x_0^2, x_1^2, x_2^2$)
- $9x_0 + 5x_1 + 11x_3 \geq 49$ (cognitive performance index from $x_0, x_1, x_3$)
- $10x_0 + 11x_1 \leq 48$ (energy stability index from $x_0, x_1$)
- $10x_0^2 + 13x_2^2 \leq 31$ (energy stability index from $x_0^2, x_2^2$)
- $10x_0 + 12x_3 \leq 60$ (energy stability index from $x_0, x_3$)
- $11x_1 + 13x_2 \leq 42$ (energy stability index from $x_1, x_2$)
- $10x_0 + 11x_1 + 13x_2 + 12x_3 \leq 42$ (energy stability index from all)
- $9x_0^2 + 11x_3^2 \leq 194$ (cognitive performance index from $x_0^2, x_3^2$)
- $9x_0 + 5x_1 \leq 228$ (cognitive performance index from $x_0, x_1$)
- $9x_0^2 + 10x_2^2 \leq 195$ (cognitive performance index from $x_0^2, x_2^2$)
- $5x_1^2 + 10x_2^2 + 11x_3^2 \leq 194$ (cognitive performance index from $x_1^2, x_2^2, x_3^2$)
- $9x_0 + 5x_1 + 11x_3 \leq 141$ (cognitive performance index from $x_0, x_1, x_3$)
- $9x_0 + 5x_1 + 10x_2 + 11x_3 \leq 141$ (cognitive performance index from all)

## 3: Specify variable bounds and types
- $x_0$ is continuous
- $x_1$ is continuous
- $x_2$ is integer
- $x_3$ is integer

## 4: Implement the problem in Gurobi
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x0")  # milligrams of vitamin B7
    x1 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="x1")  # milligrams of vitamin B12
    x2 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER, name="x2")  # milligrams of zinc
    x3 = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, vtype=gurobi.GRB.INTEGER, name="x3")  # milligrams of vitamin B2

    # Objective function
    model.setObjective(7.67 * x0**2 + 1.72 * x0 * x1 + 8.57 * x0 * x3 + 8.81 * x1**2 + 2.9 * x1 * x2 + 7.6 * x2**2 + 1.36 * x2 * x3 + 5.69 * x0 + 5.34 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(10 * x0 + 11 * x1 + 13 * x2 + 12 * x3 <= 83)
    model.addConstr(9 * x0 + 5 * x1 + 10 * x2 + 11 * x3 <= 312)
    model.addConstr(11 * x1 + 13 * x2 >= 14)
    model.addConstr(11 * x1 + 12 * x3 >= 10)
    model.addConstr(10 * x0**2 + 13 * x2**2 >= 8)
    model.addConstr(13 * x2 + 12 * x3 >= 8)
    model.addConstr(10 * x0 + 11 * x1 >= 12)
    model.addConstr(9 * x0 + 5 * x1 + 10 * x2 >= 67)
    model.addConstr(9 * x0 + 5 * x1 + 11 * x3 >= 67)
    model.addConstr(9 * x0**2 + 5 * x1**2 + 10 * x2**2 >= 49)
    model.addConstr(9 * x0 + 5 * x1 + 11 * x3 >= 49)
    model.addConstr(10 * x0 + 11 * x1 <= 48)
    model.addConstr(10 * x0**2 + 13 * x2**2 <= 31)
    model.addConstr(10 * x0 + 12 * x3 <= 60)
    model.addConstr(11 * x1 + 13 * x2 <= 42)
    model.addConstr(10 * x0 + 11 * x1 + 13 * x2 + 12 * x3 <= 42)
    model.addConstr(9 * x0**2 + 11 * x2**2 <= 194)
    model.addConstr(9 * x0 + 5 * x1 <= 228)
    model.addConstr(9 * x0**2 + 10 * x2**2 <= 195)
    model.addConstr(5 * x1**2 + 10 * x2**2 + 11 * x3**2 <= 194)
    model.addConstr(9 * x0 + 5 * x1 + 11 * x3 <= 141)
    model.addConstr(9 * x0 + 5 * x1 + 10 * x2 + 11 * x3 <= 141)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"x0: {x0.varValue}")
        print(f"x1: {x1.varValue}")
        print(f"x2: {x2.varValue}")
        print(f"x3: {x3.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```