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 define the symbolic variables as follows:
- $x_1$: The number of talent show ads purchased.
- $x_2$: The number of global news commercials purchased.

The objective is to minimize the total cost of advertising. Given that each talent show ad costs $80,000 and each global news commercial costs $30,000, the objective function can be represented as:
\[ \text{Minimize:} \quad 80,000x_1 + 30,000x_2 \]

The constraints are based on the requirements to reach at least 50 million baby boomers and 30 million millennials. Since each talent show ad is seen by 5 million baby boomers and 20 million millennials, and each global news commercial is watched by 13 million baby boomers and 7 million millennials, we have:
- For baby boomers: $5x_1 + 13x_2 \geq 50,000,000$
- For millennials: $20x_1 + 7x_2 \geq 30,000,000$

Additionally, since the number of ads cannot be negative, we have:
- $x_1 \geq 0$
- $x_2 \geq 0$

Now, let's represent this problem in the requested JSON format:

```json
{
    'sym_variables': [('x1', 'number of talent show ads'), ('x2', 'number of global news commercials')],
    'objective_function': '80000*x1 + 30000*x2',
    'constraints': ['5*x1 + 13*x2 >= 50000000', '20*x1 + 7*x2 >= 30000000', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll use the following code:

```python
from gurobipy import *

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

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

# Set the objective function
m.setObjective(80000*x1 + 30000*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(5*x1 + 13*x2 >= 50000000, "baby_boomers")
m.addConstr(20*x1 + 7*x2 >= 30000000, "millennials")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: x1 = {x1.x}, x2 = {x2.x}")
    print(f"Total cost: ${m.objVal:.2f}")
else:
    print("No optimal solution found")
```