To solve the optimization problem described, we first need to translate the given information into a mathematical formulation that can be understood by an optimization software like Gurobi. The objective is to maximize the value of \(1.79 \times \text{bowls of cereal} + 4.7 \times \text{pickles}\), subject to several constraints related to carbohydrates, sourness index, and a specific linear combination of bowls of cereal and pickles.

Let's denote:
- \(x_0\) as the quantity of bowls of cereal,
- \(x_1\) as the quantity of pickles.

The objective function to maximize is: \(1.79x_0 + 4.7x_1\).

Constraints are as follows:
1. Carbohydrates from bowls of cereal and pickles must be at least 37 grams: \(3x_0 + 3x_1 \geq 37\).
2. The total sourness index must be at least 11: \(6x_0 + 2x_1 \geq 11\).
3. A specific linear combination of bowls of cereal and pickles must be non-negative: \(10x_0 - 6x_1 \geq 0\).
4. Carbohydrates from both sources cannot exceed 46 grams (this constraint is mentioned twice but seems to be a repetition): \(3x_0 + 3x_1 \leq 46\).
5. The total sourness index must not exceed 39: \(6x_0 + 2x_1 \leq 39\).

Given that the quantities of bowls of cereal and pickles can be non-integer, we are dealing with a linear programming problem.

Here's how you could implement this in Gurobi using Python:

```python
from gurobipy import *

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

# Add variables to the model (non-integer quantities allowed)
x0 = m.addVar(name="bowls_of_cereal", lb=0, vtype=GRB.CONTINUOUS)
x1 = m.addVar(name="pickles", lb=0, vtype=GRB.CONTINUOUS)

# Set the objective function
m.setObjective(1.79*x0 + 4.7*x1, GRB.MAXIMIZE)

# Add constraints
m.addConstr(3*x0 + 3*x1 >= 37, name="min_carbohydrates")
m.addConstr(6*x0 + 2*x1 >= 11, name="min_sourness")
m.addConstr(10*x0 - 6*x1 >= 0, name="specific_linear_combination")
m.addConstr(3*x0 + 3*x1 <= 46, name="max_carbohydrates")
m.addConstr(6*x0 + 2*x1 <= 39, name="max_sourness")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"bowls_of_cereal: {x0.x}, pickles: {x1.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```

This code will solve the optimization problem and provide the quantities of bowls of cereal and pickles that maximize the given objective function while satisfying all constraints.