To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the given information.

Let's denote:
- $x_1$ as the number of units of Piano A sold.
- $x_2$ as the number of units of Piano B sold.

The objective is to minimize the total cost incurred by TomMusic while attracting at least 250 customers daily. The costs associated with each piano model are as follows:
- Cost per unit of Piano A: $20 (purchase) + $12 (other costs) = $32.
- Cost per unit of Piano B: $15 (purchase) + $4 (other costs) = $19.

The objective function to minimize is the total cost, which can be represented as:
\[ 32x_1 + 19x_2 \]

The constraints are:
1. Attract at least 250 customers daily:
\[ 25x_1 + 10x_2 \geq 250 \]
2. Maximum daily budget of $450 for purchasing the pianos:
\[ 20x_1 + 15x_2 \leq 450 \]
3. Non-negativity constraints (since the number of units cannot be negative):
\[ x_1 \geq 0, x_2 \geq 0 \]

In symbolic notation with natural language objects, we have:
```json
{
    'sym_variables': [('x1', 'Piano A units'), ('x2', 'Piano B units')],
    'objective_function': '32*x1 + 19*x2',
    'constraints': ['25*x1 + 10*x2 >= 250', '20*x1 + 15*x2 <= 450', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem using Gurobi in Python, we use the following code:
```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Piano_A_Units")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Piano_B_Units")

# Set objective function
m.setObjective(32*x1 + 19*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(25*x1 + 10*x2 >= 250, "Attract_at_least_250_customers")
m.addConstr(20*x1 + 15*x2 <= 450, "Daily_budget_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Piano A units: {x1.x}")
    print(f"Piano B units: {x2.x}")
    print(f"Total cost: {m.objVal}")
else:
    print("No optimal solution found")
```