To solve the given optimization problem, 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 problem statement.

Let's define:
- $x_1$ as the number of pairs of skis to buy and sell.
- $x_2$ as the number of snowboards to buy and sell.

The objective is to maximize profit. Each pair of skis sold gives a profit of $200, and each snowboard sold gives a profit of $175. Therefore, the objective function can be represented as:
\[ \text{Maximize} \quad 200x_1 + 175x_2 \]

Now, let's list out the constraints based on the problem description:
1. The store has a budget of $20,000. Each pair of skis costs $500, and each snowboard costs $400. So, the cost constraint is:
\[ 500x_1 + 400x_2 \leq 20000 \]
2. At least 10 but at most 30 pairs of skis will be sold:
\[ 10 \leq x_1 \leq 30 \]
3. The number of snowboards sold is at most a half the number of skis sold:
\[ x_2 \leq \frac{1}{2}x_1 \]

All variables must be non-negative since they represent quantities of items.

The symbolic representation in JSON format is:

```json
{
  'sym_variables': [('x1', 'number of pairs of skis'), ('x2', 'number of snowboards')],
  'objective_function': '200*x1 + 175*x2',
  'constraints': ['500*x1 + 400*x2 <= 20000', '10 <= x1 <= 30', 'x2 <= 0.5*x1', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's write the Gurobi code in Python to solve this optimization problem:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name='x1', lb=10, ub=30, vtype=GRB.INTEGER)
x2 = m.addVar(name='x2', lb=0, vtype=GRB.INTEGER)

# Set the objective function
m.setObjective(200*x1 + 175*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(500*x1 + 400*x2 <= 20000, name='budget_constraint')
m.addConstr(x2 <= 0.5*x1, name='snowboard_limit')

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of pairs of skis: {x1.x}")
    print(f"Number of snowboards: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```