## Problem Description and Formulation

The problem requires minimizing the objective function: \(2 \times (\text{milligrams of vitamin B4})^2 + 5 \times (\text{milligrams of vitamin B4})\), subject to several constraints involving "milligrams of vitamin B4" and "milligrams of vitamin D".

Let's denote:
- \(x_0\) as "milligrams of vitamin B4"
- \(x_1\) as "milligrams of vitamin D"

## Constraints

1. **Immune Support Index of \(x_0\) and \(x_1\)**: Given as \(r0\) with \(x_0 = 1\) and \(x_1 = 8\).
2. **Constraint 1**: \(x_0^2 + x_1^2 \geq 16\)
3. **Constraint 2**: \(x_0 + x_1 \geq 16\)
4. **Constraint 3**: \(10x_0 - 10x_1 \geq 0\)
5. **Constraint 4**: \(x_0 + x_1 \leq 19\)

## Objective Function

Minimize: \(2x_0^2 + 5x_0\)

## Gurobi Code Formulation

```python
import gurobi

def optimize_vitamins():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="milligrams_of_vitamin_B4", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x1 = model.addVar(name="milligrams_of_vitamin_D", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

    # Objective function
    model.setObjective(2*x0**2 + 5*x0, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x0**2 + x1**2 >= 16, name="constraint_1")
    model.addConstr(x0 + x1 >= 16, name="constraint_2")
    model.addConstr(10*x0 - 10*x1 >= 0, name="constraint_3")
    model.addConstr(x0 + x1 <= 19, name="constraint_4")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Milligrams of vitamin B4: {x0.varValue}")
        print(f"Milligrams of vitamin D: {x1.varValue}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_vitamins()
```