## Problem Description and Formulation

The startup needs to hire computer engineers and software engineers to minimize labor costs while meeting certain requirements. Let's denote the number of computer engineers as \(C\) and the number of software engineers as \(S\).

The requirements are:
1. The total number of engineers (\(C + S\)) must be at least 50.
2. The number of computer engineers (\(C\)) must be at least 10.
3. The number of software engineers (\(S\)) should be at least twice the number of computer engineers (\(S \geq 2C\)).

The objective is to minimize the total labor cost, given that computer engineers earn $1000 a week and software engineers earn $800 a week. The total labor cost (\(TC\)) can be represented as:
\[TC = 1000C + 800S\]

## Converting to a Linear Programming Problem

The problem can be formulated as a linear programming problem:
- **Objective Function:** Minimize \(TC = 1000C + 800S\)
- **Constraints:**
  1. \(C + S \geq 50\)
  2. \(C \geq 10\)
  3. \(S \geq 2C\)
  4. \(C \geq 0\) and \(S \geq 0\) (Non-negativity constraints, as the number of engineers cannot be negative)

## Gurobi Code

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

```python
import gurobi

def solve_engineer_hiring_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    C = model.addVar(lb=0, name="Computer_Engineers")  # Number of computer engineers
    S = model.addVar(lb=0, name="Software_Engineers")  # Number of software engineers

    # Objective function: Minimize total labor cost
    model.setObjective(1000*C + 800*S, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(C + S >= 50, name="Total_Engineers")  # At least 50 engineers in total
    model.addConstr(C >= 10, name="Min_Computer_Engineers")  # At least 10 computer engineers
    model.addConstr(S >= 2*C, name="Software_to_Computer_Ratio")  # Software engineers at least twice computer engineers

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal Solution:")
        print(f"Number of Computer Engineers: {C.varValue}")
        print(f"Number of Software Engineers: {S.varValue}")
        print(f"Total Labor Cost: ${model.objVal:.2f}")
    else:
        print("The model is infeasible.")

solve_engineer_hiring_problem()
```

This code defines the problem in Gurobi, solves it, and prints out the optimal solution if one exists.