To solve the optimization problem described, we first need to translate 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 sports ad commercials purchased.
- \(x_2\) as the number of TV show commercials purchased.

The objective is to minimize the total cost of the advertising campaign. The cost for each sports ad is $90,000, and for each TV show commercial, it's $20,000. Therefore, the objective function can be represented as:
\[ \text{Minimize} \quad 90,000x_1 + 20,000x_2 \]

The constraints are based on the requirements to reach at least 40 million baby boomers and 25 million millennials. Each sports ad reaches 4 million baby boomers and 18 million millennials, while each TV show commercial reaches 12 million baby boomers and 5 million millennials. Thus, we have:
- For baby boomers: \(4x_1 + 12x_2 \geq 40,000,000\)
- For millennials: \(18x_1 + 5x_2 \geq 25,000,000\)

Also, since the number of commercials cannot be negative, we have:
- \(x_1 \geq 0\)
- \(x_2 \geq 0\)

Symbolic representation in JSON format:
```json
{
    'sym_variables': [('x1', 'number of sports ad commercials'), ('x2', 'number of TV show commercials')],
    'objective_function': '90000*x1 + 20000*x2',
    'constraints': ['4*x1 + 12*x2 >= 40000000', '18*x1 + 5*x2 >= 25000000', 'x1 >= 0', 'x2 >= 0']
}
```

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

```python
from gurobipy import *

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

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

# Set the objective function
m.setObjective(90000*x1 + 20000*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(4*x1 + 12*x2 >= 40000000, name="baby_boomers_constraint")
m.addConstr(18*x1 + 5*x2 >= 25000000, name="millennials_constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of sports ad commercials: {x1.x}")
    print(f"Number of TV show commercials: {x2.x}")
    print(f"Total cost: ${90000*x1.x + 20000*x2.x:.2f}")
else:
    print("No optimal solution found.")

```