To solve this optimization problem, we will use the Gurobi Python library. The goal is to minimize the objective function `7 * hours_worked_by_George + 7 * hours_worked_by_Dale` subject to several constraints.

First, let's analyze and translate each constraint into a mathematical expression:

1. Paperwork competence rating for George: $5 \cdot hours\_worked\_by\_George$
2. Computer competence rating for George: $15 \cdot hours\_worked\_by\_George$
3. Paperwork competence rating for Dale: $3 \cdot hours\_worked\_by\_Dale$
4. Computer competence rating for Dale: $9 \cdot hours\_worked\_by\_Dale$

Constraints:
- Total combined paperwork competence rating $\geq 14$: $5 \cdot hours\_worked\_by\_George + 3 \cdot hours\_worked\_by\_Dale \geq 14$
- Total combined computer competence rating $\geq 14$: $15 \cdot hours\_worked\_by\_George + 9 \cdot hours\_worked\_by\_Dale \geq 14$
- Constraint on George and Dale's work: $hours\_worked\_by\_George - 7 \cdot hours\_worked\_by\_Dale \geq 0$
- Total combined paperwork competence rating $\leq 45$: $5 \cdot hours\_worked\_by\_George + 3 \cdot hours\_worked\_by\_Dale \leq 45$
- Total combined computer competence rating $\leq 50$: $15 \cdot hours\_worked\_by\_George + 9 \cdot hours\_worked\_by\_Dale \leq 50$

Given that `hours_worked_by_George` and `hours_worked_by_Dale` can be fractional, we will treat them as continuous variables.

Now, let's write the Gurobi code to represent this optimization problem:

```python
from gurobipy import *

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

# Define variables
hours_worked_by_George = m.addVar(lb=0, name="hours_worked_by_George", vtype=GRB.CONTINUOUS)
hours_worked_by_Dale = m.addVar(lb=0, name="hours_worked_by_Dale", vtype=GRB.CONTINUOUS)

# Objective function
m.setObjective(7 * hours_worked_by_George + 7 * hours_worked_by_Dale, GRB.MINIMIZE)

# Constraints
m.addConstr(5 * hours_worked_by_George + 3 * hours_worked_by_Dale >= 14, name="paperwork_min")
m.addConstr(15 * hours_worked_by_George + 9 * hours_worked_by_Dale >= 14, name="computer_min")
m.addConstr(hours_worked_by_George - 7 * hours_worked_by_Dale >= 0, name="george_dale_constraint")
m.addConstr(5 * hours_worked_by_George + 3 * hours_worked_by_Dale <= 45, name="paperwork_max")
m.addConstr(15 * hours_worked_by_George + 9 * hours_worked_by_Dale <= 50, name="computer_max")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by George: {hours_worked_by_George.x}")
    print(f"Hours worked by Dale: {hours_worked_by_Dale.x}")
else:
    print("No optimal solution found.")
```