To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints.

Let's denote:
- \(x_1\) as the number of face masks purchased,
- \(x_2\) as the number of hand sanitizers purchased.

The objective is to maximize daily profit. The profit from each face mask is $1, and from each hand sanitizer is $1.5. Therefore, the objective function can be represented as:
\[ \text{Maximize:} \quad 1x_1 + 1.5x_2 \]

Given constraints are:
1. Budget constraint: The total cost of face masks and hand sanitizers must not exceed $1000. Since each face mask costs $1.5 and each hand sanitizer costs $3, we have:
\[ 1.5x_1 + 3x_2 \leq 1000 \]
2. Face masks sales constraint: At least 80 but at most 500 face masks are sold each day:
\[ 80 \leq x_1 \leq 500 \]
3. Hand sanitizers sales constraint: The number of hand sanitizers sold will be at most half the number of face masks sold:
\[ x_2 \leq 0.5x_1 \]

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

```json
{
    'sym_variables': [('x1', 'number of face masks'), ('x2', 'number of hand sanitizers')],
    'objective_function': 'Maximize: 1*x1 + 1.5*x2',
    'constraints': [
        '1.5*x1 + 3*x2 <= 1000',
        '80 <= x1 <= 500',
        'x2 <= 0.5*x1'
    ]
}
```

Next, we'll implement this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a model
m = Model("Ecommerce_Optimization")

# Define variables
x1 = m.addVar(lb=80, ub=500, vtype=GRB.INTEGER, name="face_masks")
x2 = m.addVar(lb=0, ub=250, vtype=GRB.INTEGER, name="hand_sanitizers")  # Upper bound set based on x2 <= 0.5*x1 and x1 <= 500

# Define the objective function
m.setObjective(1*x1 + 1.5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(1.5*x1 + 3*x2 <= 1000, name="budget_constraint")
m.addConstr(x2 <= 0.5*x1, name="hand_sanitizers_sales_constraint")

# Optimize the model
m.optimize()

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