## Step 1: Define the optimization problem
The problem requires minimizing an objective function subject to various constraints. The objective function to minimize is:
\[ 1.84x_0^2 + 6.46x_0x_1 + 4.61x_1^2 + 1.45x_1x_2 + 5.7x_0 \]
where $x_0$ represents milligrams of vitamin B12, $x_1$ represents grams of protein, and $x_2$ represents milligrams of zinc.

## 2: List all constraints
The constraints are given based on the indices and their upper bounds:
- $r_0$: energy stability index
- $r_1$: digestive support index
- $r_2$: cardiovascular support index
- $r_3$: kidney support index
- $r_4$: cognitive performance index

Given indices values:
- $r_0 = 19x_0 + 4x_1 + 6x_2 \geq 11$ (for $x_0, x_1$), $r_0 = 19x_0 + 6x_2 \geq 15$ (for $x_0, x_2$), and $r_0 \geq 15$ (for all)
- $r_1 = 6x_0 + 14x_1 + 8x_2 \geq 17$ (for $x_0, x_1$), $r_1 = 6x_0 + 8x_2 \geq 26$ (for $x_0, x_2$), $r_1 \geq 27$ (for all), and $r_1 \geq 27$ specifically for $x_0^2 + x_1^2 + x_2^2$
- $r_2 = 5x_0 + 2x_1 + 2x_2 \geq 30$ (for $x_0, x_2$), $r_2 = 5x_0 + 2x_1 \geq 33$ (for $x_0, x_1$), $r_2 = 2x_1 + 2x_2 \geq 24$ (for $x_1, x_2$), and $r_2 \geq 24$ (for all)
- $r_3 = 11x_0 + 20x_1 + 14x_2 \geq 14$ (for $x_0^2, x_1^2$), $r_3 = 20x_1 + 14x_2 \geq 15$ (for $x_1, x_2$), $r_3 = 11x_0 + 14x_2 \geq 24$ (for $x_0, x_2$), and $r_3 \geq 24$ (for all)
- $r_4 = 8x_0 + 2x_1 + 20x_2 \geq 21$ (for $x_0, x_1$), $r_4 = 8x_0 + 20x_2 \geq 23$ (for $x_0, x_2$), $r_4 = 2x_1 + 20x_2 \geq 16$ (for $x_1, x_2$), and $r_4 \geq 14$ (for all)

## 3: Formulate the problem in Gurobi
To model this problem using Gurobi, we first need to define the variables, the objective function, and then add all the constraints.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define variables
x0 = m.addVar(name="milligrams_of_vitamin_B12", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x1 = m.addVar(name="grams_of_protein", lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY)
x2 = m.addVar(name="milligrams_of_zinc", lb=0, ub=gp.GRB.INFINITY, vtype=gp.GRB.INTEGER)

# Objective function
m.setObjective(1.84*x0**2 + 6.46*x0*x1 + 4.61*x1**2 + 1.45*x1*x2 + 5.7*x0, gp.GRB.MINIMIZE)

# Constraints
# Individual indices
m.addConstr(19*x0 + 4*x1 + 6*x2 <= 84)
m.addConstr(6*x0 + 14*x1 + 8*x2 <= 107)
m.addConstr(5*x0 + 2*x1 + 2*x2 <= 107)
m.addConstr(11*x0 + 20*x1 + 14*x2 <= 83)
m.addConstr(8*x0 + 2*x1 + 20*x2 <= 80)

# Energy stability index constraints
m.addConstr(19*x0 + 4*x1 >= 11)
m.addConstr(19*x0 + 6*x2 >= 15)
m.addConstr(19*x0 + 4*x1 + 6*x2 >= 15)

# Digestive support index constraints
m.addConstr(6*x0 + 14*x1 >= 17)
m.addConstr(6*x0 + 8*x2 >= 26)
m.addConstr(6*x0 + 14*x1 + 8*x2 >= 27)
m.addConstr(6*x0**2 + 14*x1**2 + 8*x2**2 >= 27)

# Cardiovascular support index constraints
m.addConstr(5*x0 + 2*x2 >= 30)
m.addConstr(5*x0 + 2*x1 >= 33)
m.addConstr(2*x1 + 2*x2 >= 24)
m.addConstr(5*x0 + 2*x1 + 2*x2 >= 24)

# Kidney support index constraints
m.addConstr(11*x0**2 + 20*x1**2 >= 14)
m.addConstr(20*x1 + 14*x2 >= 15)
m.addConstr(11*x0 + 14*x2 >= 24)
m.addConstr(11*x0 + 20*x1 + 14*x2 >= 24)

# Cognitive performance index constraints
m.addConstr(8*x0 + 2*x1 >= 21)
m.addConstr(8*x0 + 20*x2 >= 23)
m.addConstr(2*x1 + 20*x2 >= 16)
m.addConstr(8*x0 + 2*x1 + 20*x2 >= 14)

# Additional constraints
m.addConstr(-5*x0**2 + 7*x1**2 >= 0)
m.addConstr(4*x1 + 6*x2 <= 70)
m.addConstr(5*x0**2 + 2*x1**2 <= 49)
m.addConstr(11*x0 + 14*x2 <= 75)
m.addConstr(2*x1**2 + 20*x2**2 <= 64)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Milligrams of vitamin B12: {x0.varValue}")
    print(f"Grams of protein: {x1.varValue}")
    print(f"Milligrams of zinc: {x2.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```

Please ensure you have Gurobi installed and properly configured in your Python environment to run this code. The output will provide the optimal values for $x_0$, $x_1$, and $x_2$ if a solution exists.