## Problem Description and Formulation

The problem requires minimizing a given objective function subject to several constraints. The objective function and constraints involve variables representing milligrams of vitamin B1, grams of protein, and grams of fiber. The constraints include bounds on the kidney support index for each variable and combinations of them.

## Objective Function

The objective function to minimize is:
\[ 7(\text{mg of B1})^2 + 9(\text{mg of B1})(\text{g of protein}) + 4(\text{mg of B1})(\text{g of fiber}) + 4(\text{g of protein})^2 + 8(\text{g of protein})(\text{g of fiber}) + 1(\text{g of fiber})^2 + 8(\text{mg of B1}) + 4(\text{g of protein}) + 1(\text{g of fiber}) \]

## Constraints

1. Kidney support index for milligrams of vitamin B1: \( 9 \)
2. Kidney support index for grams of protein: \( 7 \)
3. Kidney support index for grams of fiber: \( 10 \)
4. \( 9 + 10 \geq 16 \)
5. \( 9 + 7 \geq 19 \)
6. \( 9 + 7 + 10 \geq 19 \)
7. \( 9^2 + 7^2 + 10^2 \leq 73 \)
8. Milligrams of vitamin B1 is an integer.
9. Grams of protein is not fractional.
10. Grams of fiber can be fractional.

## Gurobi Code Formulation

Given the problem description, let's formulate the problem using Gurobi in Python.

```python
import gurobi as gp

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

# Define variables
mg_B1 = m.addVar(name="mg_B1", vtype=gp.GRB.INTEGER)  # integer
g_protein = m.addVar(name="g_protein", vtype=gp.GRB.CONTINUOUS)  # non-fractional, so continuous but will be enforced
g_fiber = m.addVar(name="g_fiber", vtype=gp.GRB.CONTINUOUS)  # can be fractional

# Objective function
m.setObjective(7 * mg_B1 ** 2 + 9 * mg_B1 * g_protein + 4 * mg_B1 * g_fiber + 
               4 * g_protein ** 2 + 8 * g_protein * g_fiber + 1 * g_fiber ** 2 + 
               8 * mg_B1 + 4 * g_protein + 1 * g_fiber, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(mg_B1 == 9, name="kidney_support_B1")
m.addConstr(g_protein == 7, name="kidney_support_protein")
m.addConstr(g_fiber == 10, name="kidney_support_fiber")

# Since mg_B1, g_protein, g_fiber have fixed values from constraints, 
# constraints 4, 5, 6 are automatically satisfied or not needed.

# Constraint 7: sum of squares of indices
m.addConstr(9 ** 2 + 7 ** 2 + 10 ** 2 <= 73, name="sum_squares_indices")

# Solve the problem
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Milligrams of vitamin B1: {mg_B1.varValue}")
    print(f"Grams of protein: {g_protein.varValue}")
    print(f"Grams of fiber: {g_fiber.varValue}")
    print(f"Objective value: {m.objVal}")
else:
    print("No optimal solution found.")
```