To solve this optimization problem, we first need to define the decision variables and the objective function. Let's denote the number of zippered hoodies as \(Z\) and the number of pullover hoodies as \(P\).

The total cost for buying \(Z\) zippered hoodies and \(P\) pullover hoodies can be expressed as \(20Z + 15P\), which must not exceed $450. The total number of hoodies sold is constrained by \(Z + P \leq 25\). 

The profit from selling \(Z\) zippered hoodies at a profit of $5 each and \(P\) pullover hoodies at a profit of $4 each can be represented as \(5Z + 4P\).

Thus, the optimization problem can be formulated as follows:

- **Objective Function:** Maximize \(5Z + 4P\)
- **Constraints:**
  - \(20Z + 15P \leq 450\) (Total budget constraint)
  - \(Z + P \leq 25\) (Total number of hoodies constraint)
  - \(Z, P \geq 0\) and are integers (Non-negativity and integer constraints)

Now, let's implement this optimization problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
m = Model("Hoodie_Optimization")

# Define the decision variables
Z = m.addVar(vtype=GRB.INTEGER, name="Zippered_Hoodies")
P = m.addVar(vtype=GRB.INTEGER, name="Pullover_Hoodies")

# Set the objective function to maximize profit
m.setObjective(5*Z + 4*P, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*Z + 15*P <= 450, "Budget_Constraint")
m.addConstr(Z + P <= 25, "Total_Hoodies_Constraint")

# Solve the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")
print(f"Profit: {m.objVal}")

```
```python
from gurobipy import *

# Create a new model
m = Model("Hoodie_Optimization")

# Define the decision variables
Z = m.addVar(vtype=GRB.INTEGER, name="Zippered_Hoodies")
P = m.addVar(vtype=GRB.INTEGER, name="Pullover_Hoodies")

# Set the objective function to maximize profit
m.setObjective(5*Z + 4*P, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*Z + 15*P <= 450, "Budget_Constraint")
m.addConstr(Z + P <= 25, "Total_Hoodies_Constraint")

# Solve the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")
print(f"Profit: {m.objVal}")

```