To solve the optimization problem described, we will use linear programming techniques. The goal is to minimize the total cost of mixing bags of everything seasoning while meeting the minimum requirements for sesame seeds, onion powder, and garlic powder in the house mix.

Let's denote:
- \(x\) as the number of ordinary bags used,
- \(y\) as the number of special bags used.

The constraints based on the ingredients are as follows:
1. Sesame seeds: \(5x + 10y \geq 50\)
2. Onion powder: \(8x + 6y \geq 60\)
3. Garlic powder: \(7x + 8y \geq 65\)

The objective function to minimize the total cost is:
\[10x + 12y\]

We will use Gurobi, a popular optimization solver, with Python as our programming language to solve this problem.

```python
from gurobipy import *

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

# Define variables
x = m.addVar(vtype=GRB.CONTINUOUS, name="ordinary_bags")
y = m.addVar(vtype=GRB.CONTINUOUS, name="special_bags")

# Define constraints
m.addConstr(5*x + 10*y >= 50, "sesame_seeds")
m.addConstr(8*x + 6*y >= 60, "onion_powder")
m.addConstr(7*x + 8*y >= 65, "garlic_powder")

# Define objective function
m.setObjective(10*x + 12*y, GRB.MINIMIZE)

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Ordinary bags: {x.x}")
    print(f"Special bags: {y.x}")
    print(f"Total cost: ${10*x.x + 12*y.x:.2f}")
else:
    print("No optimal solution found")

```
```python
from gurobipy import *

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

# Define variables
x = m.addVar(vtype=GRB.CONTINUOUS, name="ordinary_bags")
y = m.addVar(vtype=GRB.CONTINUOUS, name="special_bags")

# Define constraints
m.addConstr(5*x + 10*y >= 50, "sesame_seeds")
m.addConstr(8*x + 6*y >= 60, "onion_powder")
m.addConstr(7*x + 8*y >= 65, "garlic_powder")

# Define objective function
m.setObjective(10*x + 12*y, GRB.MINIMIZE)

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Ordinary bags: {x.x}")
    print(f"Special bags: {y.x}")
    print(f"Total cost: ${10*x.x + 12*y.x:.2f}")
else:
    print("No optimal solution found")
```