## Step 1: Define the optimization problem
The problem is to minimize the objective function: $8 \cdot (\text{grams of protein})^2 + 9 \cdot (\text{grams of protein}) \cdot (\text{mg of vitamin D}) + 5 \cdot (\text{grams of protein}) \cdot (\text{mg of vitamin B1}) + 4 \cdot (\text{mg of vitamin D}) \cdot (\text{mg of vitamin B7}) + 3 \cdot (\text{mg of vitamin B1})^2 + 6 \cdot (\text{mg of vitamin B1}) \cdot (\text{mg of vitamin B7}) + 6 \cdot (\text{mg of vitamin B1}) \cdot (\text{mg of vitamin C}) + 2 \cdot (\text{mg of vitamin D}) + 2 \cdot (\text{mg of vitamin B1}) + 4 \cdot (\text{mg of vitamin B7}) + 9 \cdot (\text{mg of vitamin C})$.

## Step 2: List all the constraints
The constraints are:
- $5 \cdot (\text{grams of protein}) \leq 172$
- $4 \cdot (\text{grams of protein}) \leq 79$
- $10 \cdot (\text{mg of vitamin D}) \leq 172$
- $1 \cdot (\text{mg of vitamin D}) \leq 79$
- $7 \cdot (\text{mg of vitamin B1}) \leq 172$
- $11 \cdot (\text{mg of vitamin B1}) \leq 79$
- $6 \cdot (\text{mg of vitamin B7}) \leq 172$
- $7 \cdot (\text{mg of vitamin B7}) \leq 79$
- $7 \cdot (\text{mg of vitamin C}) \leq 172$
- $3 \cdot (\text{mg of vitamin C}) \leq 79$
- $7 \cdot (\text{mg of vitamin B1}) + 6 \cdot (\text{mg of vitamin B7}) \geq 28$
- $(\text{grams of protein})^2 + (\text{mg of vitamin D})^2 \geq 25$
- $10 \cdot (\text{mg of vitamin D}) + 6 \cdot (\text{mg of vitamin B7}) \geq 12$
- $5 \cdot (\text{grams of protein}) + 7 \cdot (\text{mg of vitamin C}) \geq 34$
- $(\text{mg of vitamin B1})^2 + (\text{mg of vitamin B7})^2 + (\text{mg of vitamin C})^2 \geq 33$
- $(\text{grams of protein})^2 + (\text{mg of vitamin B7})^2 + (\text{mg of vitamin C})^2 \geq 33$
- $5 \cdot (\text{grams of protein}) + 7 \cdot (\text{mg of vitamin B1}) + 7 \cdot (\text{mg of vitamin C}) \geq 33$
- $(\text{mg of vitamin B1})^2 + (\text{mg of vitamin B7})^2 + (\text{mg of vitamin C})^2 \geq 33$
- $5 \cdot (\text{grams of protein}) + 6 \cdot (\text{mg of vitamin B7}) + 7 \cdot (\text{mg of vitamin C}) \geq 33$
- $5 \cdot (\text{grams of protein}) + 7 \cdot (\text{mg of vitamin B1}) + 7 \cdot (\text{mg of vitamin C}) \geq 33$
- $7 \cdot (\text{mg of vitamin B1}) + 6 \cdot (\text{mg of vitamin B7}) + 7 \cdot (\text{mg of vitamin C}) \geq 32$
- $5 \cdot (\text{grams of protein}) + 6 \cdot (\text{mg of vitamin B7}) + 7 \cdot (\text{mg of vitamin C}) \geq 32$
- $(\text{grams of protein})^2 + (\text{mg of vitamin B1})^2 + (\text{mg of vitamin C})^2 \geq 32$
- $5 \cdot (\text{grams of protein}) + 10 \cdot (\text{mg of vitamin D}) + 7 \cdot (\text{mg of vitamin B1}) + 6 \cdot (\text{mg of vitamin B7}) + 7 \cdot (\text{mg of vitamin C}) \geq 32$
- $1 \cdot (\text{mg of vitamin D}) + 11 \cdot (\text{mg of vitamin B1}) \geq 6$
- $4 \cdot (\text{grams of protein}) + 11 \cdot (\text{mg of vitamin B1}) \geq 6$
- $4 \cdot (\text{grams of protein}) + 3 \cdot (\text{mg of vitamin C}) \geq 6$
- $4 \cdot (\text{grams of protein}) + 1 \cdot (\text{mg of vitamin D}) + 11 \cdot (\text{mg of vitamin B1}) + 7 \cdot (\text{mg of vitamin B7}) + 3 \cdot (\text{mg of vitamin C}) \geq 6$
- $-9 \cdot (\text{mg of vitamin B1}) + 10 \cdot (\text{mg of vitamin B7}) \geq 0$
- $-9 \cdot (\text{grams of protein}) + 8 \cdot (\text{mg of vitamin C}) \geq 0$
- $5 \cdot (\text{grams of protein}) + 6 \cdot (\text{mg of vitamin B7}) \leq 134$
- $10 \cdot (\text{mg of vitamin D}) + 6 \cdot (\text{mg of vitamin B7}) \leq 47$
- $(\text{grams of protein})^2 + (\text{mg of vitamin B1})^2 \leq 97$
- $(\text{mg of vitamin B1})^2 + (\text{mg of vitamin B7})^2 \leq 116$
- $6 \cdot (\text{mg of vitamin B7}) + 7 \cdot (\text{mg of vitamin C}) \leq 144$
- $7 \cdot (\text{mg of vitamin B1}) + 7 \cdot (\text{mg of vitamin C}) \leq 148$
- $(\text{mg of vitamin D})^2 + (\text{mg of vitamin B1})^2 \leq 164$
- $5 \cdot (\text{grams of protein}) + 7 \cdot (\text{mg of vitamin B1}) + 7 \cdot (\text{mg of vitamin C}) \leq 48$
- $(\text{grams of protein})^2 + (\text{mg of vitamin C})^2 \leq 65$
- $4 \cdot (\text{grams of protein}) + 1 \cdot (\text{mg of vitamin D}) \leq 45$
- $4 \cdot (\text{grams of protein}) + 11 \cdot (\text{mg of vitamin B1}) \leq 37$
- $(\text{grams of protein})^2 + (\text{mg of vitamin D})^2 + (\text{mg of vitamin B1})^2 \leq 27$
- $11 \cdot (\text{mg of vitamin B1}) + 7 \cdot (\text{mg of vitamin B7}) + 3 \cdot (\text{mg of vitamin C}) \leq 53$
- $(\text{grams of protein})^2 + (\text{mg of vitamin B7})^2 + (\text{mg of vitamin C})^2 \leq 65$
- $(\text{mg of vitamin D})^2 + (\text{mg of vitamin B1})^2 + (\text{mg of vitamin C})^2 \leq 31$
- $1 \cdot (\text{mg of vitamin D}) + 11 \cdot (\text{mg of vitamin B1}) + 7 \cdot (\text{mg of vitamin B7}) \leq 21$
- $(\text{grams of protein})^2 + (\text{mg of vitamin B1})^2 + (\text{mg of vitamin B7})^2 \leq 26$
- $1 \cdot (\text{mg of vitamin D}) + 7 \cdot (\text{mg of vitamin B7}) + 3 \cdot (\text{mg of vitamin C}) \leq 42$
- $(\text{grams of protein})^2 + (\text{mg of vitamin D})^2 + (\text{mg of vitamin C})^2 \leq 16$

## Step 3: Implement the optimization problem using Gurobi
We will use the Gurobi Python API to model and solve this optimization problem.

```python
import gurobi as gp

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

# Define the variables
protein = m.addVar(lb=0, name="protein")  # grams of protein
vitamin_D = m.addVar(lb=0, name="vitamin_D")  # milligrams of vitamin D
vitamin_B1 = m.addVar(lb=0, name="vitamin_B1")  # milligrams of vitamin B1
vitamin_B7 = m.addVar(lb=0, name="vitamin_B7")  # milligrams of vitamin B7
vitamin_C = m.addVar(lb=0, integer=True, name="vitamin_C")  # milligrams of vitamin C

# Define the objective function
m.setObjective(8 * protein**2 + 9 * protein * vitamin_D + 5 * protein * vitamin_B1 + 
               4 * vitamin_D * vitamin_B7 + 3 * vitamin_B1**2 + 6 * vitamin_B1 * vitamin_B7 + 
               6 * vitamin_B1 * vitamin_C + 2 * vitamin_D + 2 * vitamin_B1 + 
               4 * vitamin_B7 + 9 * vitamin_C, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(5 * protein <= 172)
m.addConstr(4 * protein <= 79)
m.addConstr(10 * vitamin_D <= 172)
m.addConstr(vitamin_D <= 79)
m.addConstr(7 * vitamin_B1 <= 172)
m.addConstr(11 * vitamin_B1 <= 79)
m.addConstr(6 * vitamin_B7 <= 172)
m.addConstr(7 * vitamin_B7 <= 79)
m.addConstr(7 * vitamin_C <= 172)
m.addConstr(3 * vitamin_C <= 79)

m.addConstr(7 * vitamin_B1 + 6 * vitamin_B7 >= 28)
m.addConstr(protein**2 + vitamin_D**2 >= 25)
m.addConstr(10 * vitamin_D + 6 * vitamin_B7 >= 12)
m.addConstr(5 * protein + 7 * vitamin_C >= 34)
m.addConstr(vitamin_B1**2 + vitamin_B7**2 + vitamin_C**2 >= 33)
m.addConstr(protein**2 + vitamin_B7**2 + vitamin_C**2 >= 33)
m.addConstr(5 * protein + 7 * vitamin_B1 + 7 * vitamin_C >= 33)
m.addConstr(vitamin_B1**2 + vitamin_B7**2 + vitamin_C**2 >= 33)
m.addConstr(5 * protein + 6 * vitamin_B7 + 7 * vitamin_C >= 33)
m.addConstr(5 * protein + 7 * vitamin_B1 + 7 * vitamin_C >= 33)
m.addConstr(7 * vitamin_B1 + 6 * vitamin_B7 + 7 * vitamin_C >= 32)
m.addConstr(5 * protein + 6 * vitamin_B7 + 7 * vitamin_C >= 32)
m.addConstr(protein**2 + vitamin_B1**2 + vitamin_C**2 >= 32)
m.addConstr(5 * protein + 10 * vitamin_D + 7 * vitamin_B1 + 6 * vitamin_B7 + 7 * vitamin_C >= 32)

m.addConstr(vitamin_D + 11 * vitamin_B1 >= 6)
m.addConstr(4 * protein + 11 * vitamin_B1 >= 6)
m.addConstr(4 * protein + 3 * vitamin_C >= 6)
m.addConstr(4 * protein + vitamin_D + 11 * vitamin_B1 + 7 * vitamin_B7 + 3 * vitamin_C >= 6)

m.addConstr(-9 * vitamin_B1 + 10 * vitamin_B7 >= 0)
m.addConstr(-9 * protein + 8 * vitamin_C >= 0)

m.addConstr(5 * protein + 6 * vitamin_B7 <= 134)
m.addConstr(10 * vitamin_D + 6 * vitamin_B7 <= 47)
m.addConstr(protein**2 + vitamin_B1**2 <= 97)
m.addConstr(vitamin_B1**2 + vitamin_B7**2 <= 116)
m.addConstr(6 * vitamin_B7 + 7 * vitamin_C <= 144)
m.addConstr(7 * vitamin_B1 + 7 * vitamin_C <= 148)
m.addConstr(vitamin_D**2 + vitamin_B1**2 <= 164)
m.addConstr(5 * protein + 7 * vitamin_B1 + 7 * vitamin_C <= 48)

m.addConstr(protein**2 + vitamin_C**2 <= 65)
m.addConstr(4 * protein + vitamin_D <= 45)
m.addConstr(4 * protein + 11 * vitamin_B1 <= 37)
m.addConstr(protein**2 + vitamin_D**2 + vitamin_B1**2 <= 27)
m.addConstr(11 * vitamin_B1 + 7 * vitamin_B7 + 3 * vitamin_C <= 53)
m.addConstr(protein**2 + vitamin_B7**2 + vitamin_C**2 <= 65)
m.addConstr(vitamin_D**2 + vitamin_B1**2 + vitamin_C**2 <= 31)
m.addConstr(vitamin_D + 11 * vitamin_B1 + 7 * vitamin_B7 <= 21)
m.addConstr(protein**2 + vitamin_B1**2 + vitamin_B7**2 <= 26)
m.addConstr(vitamin_D + 7 * vitamin_B7 + 3 * vitamin_C <= 42)
m.addConstr(protein**2 + vitamin_D**2 + vitamin_C**2 <= 16)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Protein: {protein.varValue}")
    print(f"Vitamin D: {vitamin_D.varValue}")
    print(f"Vitamin B1: {vitamin_B1.varValue}")
    print(f"Vitamin B7: {vitamin_B7.varValue}")
    print(f"Vitamin C: {vitamin_C.varValue}")
else:
    print("No optimal solution found.")
```