To solve the given optimization problem, we first need to identify the decision variables, objective function, and constraints.

- Decision Variables: Let's denote the number of hot dogs as `x0`, the number of cornichons as `x1`, and the number of strawberries as `x2`.
- Objective Function: The goal is to minimize `7*x0 + 6*x1 + 8*x2`.
- Constraints:
  - Protein constraints from each food item.
  - Fat constraints from each combination of food items.
  - Mixed constraints involving both protein and fat, or specific combinations of food items.

Given the complexity of the problem with multiple constraints and the objective function, we can directly translate these into Gurobi code. We'll use the `gurobipy` library in Python to model this linear programming problem.

Here is how you could implement it:

```python
from gurobipy import *

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

# Define the decision variables
x0 = m.addVar(vtype=GRB.CONTINUOUS, name="hot_dogs")
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="cornichons")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="strawberries")

# Objective function
m.setObjective(7*x0 + 6*x1 + 8*x2, GRB.MINIMIZE)

# Constraints
# Protein constraints
m.addConstr(22*x0 + 9*x2 >= 53, "protein_hot_dogs_strawberries")
m.addConstr(3*x1 + 9*x2 >= 34, "protein_cornichons_strawberries")
m.addConstr(22*x0 + 3*x1 >= 35, "protein_hot_dogs_cornichons")
m.addConstr(22*x0 + 3*x1 + 9*x2 >= 35, "protein_all")

# Fat constraints
m.addConstr(23*x0 + 19*x1 >= 26, "fat_hot_dogs_cornichons")
m.addConstr(23*x0 + 7*x2 >= 46, "fat_hot_dogs_strawberries")
m.addConstr(23*x0 + 19*x1 + 7*x2 >= 46, "fat_all")

# Mixed constraints
m.addConstr(7*x0 - 6*x2 >= 0, "mixed_hot_dogs_strawberries")
m.addConstr(5*x0 - 3*x1 >= 0, "mixed_hot_dogs_cornichons")

# Upper bound protein constraints
m.addConstr(22*x0 + 9*x2 <= 127, "upper_protein_hot_dogs_strawberries")
m.addConstr(3*x1 + 9*x2 <= 94, "upper_protein_cornichons_strawberries")
m.addConstr(22*x0 + 3*x1 + 9*x2 <= 116, "upper_protein_all")

# Upper bound fat constraint
m.addConstr(23*x0 + 19*x1 + 7*x2 <= 96, "upper_fat_all")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hot dogs: {x0.x}")
    print(f"Cornichons: {x1.x}")
    print(f"Strawberries: {x2.x}")
else:
    print("No optimal solution found")
```