## Step 1: Define the optimization problem
The problem is to minimize the objective function: $3x_0^2 + 2x_0x_2 + 7x_0x_3 + x_1^2 + 7x_1x_3 + x_2^2 + 4x_2x_3 + 3x_3^2 + 3x_0 + 3x_1 + 7x_2 + 6x_3$, where $x_0$ represents milligrams of vitamin B3, $x_1$ represents milligrams of vitamin B6, $x_2$ represents milligrams of vitamin B9, and $x_3$ represents milligrams of vitamin B5.

## Step 2: Define the constraints
The constraints are:
- $9x_0 \leq 118$
- $4x_1 \leq 118$
- $2x_2 \leq 118$
- $19x_3 \leq 118$
- $4x_1 + 19x_3 \geq 19$
- $16x_1^2 + 4x_2^2 \geq 13 * 118^2$ is not needed as it was incorrectly transcribed, instead we use $4x_1 + 2x_2 \geq 13$ seems not correct based on original, use $4x_1^2 + 2x_2^2 \geq 13$ 
- $9x_0 + 4x_1 \geq 11$
- $9x_0 + 19x_3 \geq 18$
- $4x_1 + 2x_2 + 19x_3 \geq 25$
- $9x_0 + 2x_2 + 19x_3 \geq 25$
- $9x_0 + 4x_1 + 19x_3 \geq 25$
- $4x_1 + 2x_2 + 19x_3 \geq 21$
- $9x_0 + 2x_2 + 19x_3 \geq 21$
- $9x_0 + 4x_1 + 19x_3 \geq 21$
- $4x_1 + 2x_2 + 19x_3 \geq 23$
- $9x_0 + 2x_2 + 19x_3 \geq 23$
- $9x_0 + 4x_1 + 19x_3 \geq 23$
- $9x_0 + 4x_1 + 2x_2 + 19x_3 \geq 23$
- $81x_0^2 - 7x_3^2 \geq 0$
- $8x_0 - 8x_1 \geq 0$
- $9x_0 + 2x_2 \leq 59$

## 3: Implement the problem in Gurobi
We will use Gurobi to solve this optimization problem.

```python
import gurobi as gp
from gurobi import GRB

# Create a new model
m = gp.Model("vitamin")

# Define the variables
x0 = m.addVar(lb=-gp.GRB.INF, ub=gp.GRB.INF, name="x0") # milligrams of vitamin B3
x1 = m.addVar(lb=-gp.GRB.INF, ub=gp.GRB.INF, name="x1") # milligrams of vitamin B6
x2 = m.addVar(lb=0, ub=gp.GRB.INF, integrality=1, name="x2") # milligrams of vitamin B9, integer
x3 = m.addVar(lb=0, ub=gp.GRB.INF, integrality=1, name="x3") # milligrams of vitamin B5, integer

# Objective function
m.setObjective(3*x0**2 + 2*x0*x2 + 7*x0*x3 + x1**2 + 7*x1*x3 + x2**2 + 4*x2*x3 + 3*x3**2 + 3*x0 + 3*x1 + 7*x2 + 6*x3, GRB.MINIMIZE)

# Constraints
m.addConstr(9 * x0 <= 118)
m.addConstr(4 * x1 <= 118)
m.addConstr(2 * x2 <= 118)
m.addConstr(19 * x3 <= 118)

m.addConstr(4 * x1 + 19 * x3 >= 19)
m.addConstr(x1**2 * 16 + x2**2 * 4 >= 13) 
m.addConstr(9 * x0 + 4 * x1 >= 11)
m.addConstr(9 * x0 + 19 * x3 >= 18)
m.addConstr(4 * x1 + 2 * x2 + 19 * x3 >= 25)
m.addConstr(9 * x0 + 2 * x2 + 19 * x3 >= 25)
m.addConstr(9 * x0 + 4 * x1 + 19 * x3 >= 25)
m.addConstr(4 * x1 + 2 * x2 + 19 * x3 >= 21)
m.addConstr(9 * x0 + 2 * x2 + 19 * x3 >= 21)
m.addConstr(9 * x0 + 4 * x1 + 19 * x3 >= 21)
m.addConstr(4 * x1 + 2 * x2 + 19 * x3 >= 23)
m.addConstr(9 * x0 + 2 * x2 + 19 * x3 >= 23)
m.addConstr(9 * x0 + 4 * x1 + 19 * x3 >= 23)
m.addConstr(9 * x0 + 4 * x1 + 2 * x2 + 19 * x3 >= 23)

m.addConstr(81 * x0**2 - 7 * x3**2 >= 0)
m.addConstr(8 * x0 - 8 * x1 >= 0)
m.addConstr(9 * x0 + 2 * x2 <= 59)

# Solve the model
m.optimize()

if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print("x0:", x0.varValue)
    print("x1:", x1.varValue)
    print("x2:", x2.varValue)
    print("x3:", x3.varValue)
    print("Objective:", m.objVal)
else:
    print("No optimal solution found.")
```