To solve the given optimization problem, we first need to define the variables and the objective function, and then list all the constraints.

### Symbolic Representation

Let's denote:
- $x_0$ as 'air defense batteries'
- $x_1$ as 'mechanized infantry companies'
- $x_2$ as 'signal platoons'
- $x_3$ as 'reconnaissance troops'

The objective function to minimize is:
\[ 4x_0^2 + 6x_0x_2 + 7x_0x_3 + 5x_1^2 + 7x_1x_2 + 3x_1 + 9x_2^2 + x_3 \]

The constraints are as follows:

### Constraints

1. $4x_0 + x_1 + 3x_2 + 3x_3 \leq 135$ (deployment weight)
2. $3x_0 + 2x_1 + 5x_2 + x_3 \geq 124$ (fun factor)
3. $2x_0 + 5x_1 + x_2 + 3x_3 \leq 115$ (fuel demand)
4. $x_1^2 + x_3^2 \geq 12$ 
5. $4x_0 + 3x_2 \geq 22$ 
6. $4x_0 + x_1 + 3x_2 \geq 31$ 
7. $4x_0 + 3x_2 + 3x_3 \geq 31$ 
8. $x_1^2 + x_2^2 + x_3^2 \geq 31$ 
9. $4x_0 + x_1 + 3x_2 \geq 18$ 
10. $4x_0 + 3x_2 + 3x_3 \geq 18$ 
11. $x_1 + 3x_2 + 3x_3 \geq 18$ 
12. $16x_0^2 + x_1^2 + 9x_2^2 \geq 27$ 
13. $4x_0 + 3x_2 + 3x_3 \geq 27$ 
14. $x_1 + 3x_2 + 3x_3 \geq 27$ 
15. $4x_0 + x_1 + 3x_2 + 3x_3 \geq 27$ 
16. $3x_0 + 5x_2 \geq 12$ 
17. $2x_1 + x_3 \geq 26$ 
18. $3x_0 + 2x_1 + 5x_2 \geq 20$ 
19. $x_1^2 + x_2^2 + x_3^2 \geq 20$ 
20. $9x_0^2 + x_2^2 + x_3^2 \geq 20$ 
21. $3x_0 + 2x_1 + 5x_2 \geq 19$ 
22. $x_1^2 + x_2^2 + x_3^2 \geq 19$ 
23. $3x_0 + 5x_2 + x_3 \geq 19$ 
24. $9x_0^2 + x_1^2 + 9x_2^2 \geq 29$ 
25. $x_1 + 5x_2 + x_3 \geq 29$ 
26. $3x_0 + 5x_2 + x_3 \geq 29$ 
27. $3x_0 + 2x_1 + 5x_2 + x_3 \geq 29$ 
28. $2x_1^2 + 3x_3^2 \geq 28$ 
29. $x_1^2 + 9x_2^2 \geq 24$ 
30. $2x_0 + x_2 \geq 26$ 
31. $4x_0^2 + x_1^2 \geq 25$ 
32. $x_2 + 3x_3 \geq 21$ 
33. $2x_0 + 5x_1 + x_2 + 3x_3 \geq 21$ 
34. $3x_2^2 - 4x_3^2 \geq 0$ 
35. $-3x_0 + 3x_2 \geq 0$ 
36. $4x_0 + x_1 + 3x_2 \leq 113$ 
37. $9x_0^2 + x_1^2 \leq 35$ 
38. $5x_2 + x_3 \leq 59$ 
39. $3x_0 + x_3 \leq 69$ 
40. $2x_0 + x_2 + 3x_3 \leq 65$

### Gurobi Code

```python
import gurobi

def optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name="x0", vtype=gurobi.GRB.INTEGER)  # air defense batteries
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # mechanized infantry companies
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # signal platoons
    x3 = model.addVar(name="x3", vtype=gurobi.GRB.INTEGER)  # reconnaissance troops

    # Objective function
    model.setObjective(4 * x0**2 + 6 * x0 * x2 + 7 * x0 * x3 + 5 * x1**2 + 7 * x1 * x2 + 3 * x1 + 9 * x2**2 + x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(4 * x0 + x1 + 3 * x2 + 3 * x3 <= 135)
    model.addConstr(3 * x0 + 2 * x1 + 5 * x2 + x3 >= 124)
    model.addConstr(2 * x0 + 5 * x1 + x2 + 3 * x3 <= 115)
    model.addConstr(x1**2 + x3**2 >= 12)
    model.addConstr(4 * x0 + 3 * x2 >= 22)
    model.addConstr(4 * x0 + x1 + 3 * x2 >= 31)
    model.addConstr(4 * x0 + 3 * x2 + 3 * x3 >= 31)
    model.addConstr(x1**2 + x2**2 + x3**2 >= 31)
    model.addConstr(4 * x0 + x1 + 3 * x2 >= 18)
    model.addConstr(4 * x0 + 3 * x2 + 3 * x3 >= 18)
    model.addConstr(x1 + 3 * x2 + 3 * x3 >= 18)
    model.addConstr(16 * x0**2 + x1**2 + 9 * x2**2 >= 27)
    model.addConstr(4 * x0 + 3 * x2 + 3 * x3 >= 27)
    model.addConstr(x1 + 3 * x2 + 3 * x3 >= 27)
    model.addConstr(4 * x0 + x1 + 3 * x2 + 3 * x3 >= 27)
    model.addConstr(3 * x0 + 5 * x2 >= 12)
    model.addConstr(2 * x1 + x3 >= 26)
    model.addConstr(3 * x0 + 2 * x1 + 5 * x2 >= 20)
    model.addConstr(x1**2 + x2**2 + x3**2 >= 20)
    model.addConstr(9 * x0**2 + x2**2 + x3**2 >= 20)
    model.addConstr(3 * x0 + 2 * x1 + 5 * x2 >= 19)
    model.addConstr(x1**2 + x2**2 + x3**2 >= 19)
    model.addConstr(3 * x0 + 5 * x2 + x3 >= 19)
    model.addConstr(9 * x0**2 + x1**2 + 9 * x2**2 >= 29)
    model.addConstr(x1 + 5 * x2 + x3 >= 29)
    model.addConstr(3 * x0 + 5 * x2 + x3 >= 29)
    model.addConstr(3 * x0 + 2 * x1 + 5 * x2 + x3 >= 29)
    model.addConstr(2 * x1**2 + 3 * x3**2 >= 28)
    model.addConstr(x1**2 + 9 * x2**2 >= 24)
    model.addConstr(2 * x0 + x2 >= 26)
    model.addConstr(4 * x0**2 + x1**2 >= 25)
    model.addConstr(x2 + 3 * x3 >= 21)
    model.addConstr(2 * x0 + 5 * x1 + x2 + 3 * x3 >= 21)
    model.addConstr(3 * x2**2 - 4 * x3**2 >= 0)
    model.addConstr(-3 * x0 + 3 * x2 >= 0)
    model.addConstr(4 * x0 + x1 + 3 * x2 <= 113)
    model.addConstr(9 * x0**2 + x1**2 <= 35)
    model.addConstr(5 * x2 + x3 <= 59)
    model.addConstr(3 * x0 + x3 <= 69)
    model.addConstr(2 * x0 + x2 + 3 * x3 <= 65)

    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"x0: {model.x[0].x}, x1: {model.x[1].x}, x2: {model.x[2].x}, x3: {model.x[3].x}")
        print(f"Objective: {model.objVal}")
    else:
        print("No optimal solution found.")

optimization_problem()
```