To solve this problem, we first need to understand the constraints and the objective function. The problem involves several variables: 'cherry pies', 'knishes', 'chicken thighs', 'fruit salads', 'bowls of instant ramen', and 'hot dogs'. Each of these variables has certain attributes and constraints associated with them, particularly in relation to an 'umami index'.

The objective function to minimize is:
\[ 4(\text{cherry pies})^2 + 4(\text{cherry pies})(\text{knishes}) + 9(\text{knishes})^2 + 8(\text{knishes})(\text{fruit salads}) + 3(\text{knishes})(\text{bowls of instant ramen}) + 8(\text{knishes})(\text{hot dogs}) + 9(\text{chicken thighs})^2 + (\text{chicken thighs})(\text{fruit salads}) + 6(\text{fruit salads})(\text{bowls of instant ramen}) + 2(\text{hot dogs})^2 + 4(\text{cherry pies}) + 5(\text{knishes}) + 4(\text{chicken thighs}) + 8(\text{hot dogs}) \]

Subject to numerous constraints, including umami index limits and requirements for integer or non-integer solutions for certain variables.

Here is a Python code using Gurobi to solve this optimization problem:

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
cherry_pies = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="cherry_pies")
knishes = m.addVar(lb=0, ub=gp.GRB.INFINITY, integrality=1, name="knishes")
chicken_thighs = m.addVar(lb=0, ub=gp.GRB.INFINITY, integrality=1, name="chicken_thighs")
fruit_salads = m.addVar(lb=0, ub=gp.GRB.INFINITY, integrality=1, name="fruit_salads")
bowls_of_instant_ramen = m.addVar(lb=0, ub=gp.GRB.INFINITY, integrality=1, name="bowls_of_instant_ramen")
hot_dogs = m.addVar(lb=-gp.GRB.INFINITY, ub=gp.GRB.INFINITY, name="hot_dogs")

# Objective function
m.setObjective(4*cherry_pies**2 + 4*cherry_pies*knishes + 9*knishes**2 + 8*knishes*fruit_salads + 
               3*knishes*bowls_of_instant_ramen + 8*knishes*hot_dogs + 9*chicken_thighs**2 + 
               chicken_thighs*fruit_salads + 6*fruit_salads*bowls_of_instant_ramen + 2*hot_dogs**2 + 
               4*cherry_pies + 5*knishes + 4*chicken_thighs + 8*hot_dogs, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(5 * cherry_pies + 5 * knishes + 3 * chicken_thighs + 5 * fruit_salads + 4 * bowls_of_instant_ramen + 4 * hot_dogs <= 51)
m.addConstr(5 * knishes + 4 * hot_dogs >= 3)
m.addConstr(5 * cherry_pies + 5 * knishes >= 2)
m.addConstr(5 * cherry_pies + 4 * bowls_of_instant_ramen + 4 * hot_dogs >= 7)
m.addConstr(5 * knishes + 3 * chicken_thighs + 5 * fruit_salads >= 7)
m.addConstr(5 * cherry_pies + 5 * knishes + 3 * chicken_thighs >= 7)
m.addConstr(cherry_pies**2 + knishes**2 + hot_dogs**2 >= 7)
m.addConstr(cherry_pies + knishes + bowls_of_instant_ramen >= 7)
m.addConstr(chicken_thighs + fruit_salads + bowls_of_instant_ramen >= 7)
m.addConstr(cherry_pies + fruit_salads + hot_dogs >= 7)
m.addConstr(fruit_salads**2 + bowls_of_instant_ramen**2 + hot_dogs**2 >= 7)
m.addConstr(knishes**2 + fruit_salads**2 + hot_dogs**2 >= 7)
m.addConstr(chicken_thighs**2 + bowls_of_instant_ramen**2 + hot_dogs**2 >= 7)
m.addConstr(cherry_pies + bowls_of_instant_ramen + hot_dogs >= 7)
m.addConstr(knishes + chicken_thighs + fruit_salads >= 7)
m.addConstr(cherry_pies**2 + knishes**2 + chicken_thighs**2 >= 7)
m.addConstr(cherry_pies + knishes + hot_dogs >= 7)
m.addConstr(cherry_pies + knishes + bowls_of_instant_ramen >= 7)
m.addConstr(chicken_thighs**2 + fruit_salads**2 + bowls_of_instant_ramen**2 >= 7)
m.addConstr(cherry_pies**2 + fruit_salads**2 + hot_dogs**2 >= 7)
m.addConstr(fruit_salads + bowls_of_instant_ramen + hot_dogs >= 7)
m.addConstr(knishes + fruit_salads + hot_dogs >= 7)
m.addConstr(chicken_thighs + bowls_of_instant_ramen + hot_dogs >= 7)
m.addConstr(cherry_pies**2 + bowls_of_instant_ramen**2 + hot_dogs**2 >= 6)
m.addConstr(knishes + chicken_thighs + fruit_salads >= 6)
m.addConstr(cherry_pies + knishes + chicken_thighs >= 6)
m.addConstr(cherry_pies + knishes + hot_dogs >= 6)
m.addConstr(cherry_pies + knishes + bowls_of_instant_ramen >= 6)
m.addConstr(chicken_thighs + fruit_salads + bowls_of_instant_ramen >= 6)
m.addConstr(cherry_pies**2 + fruit_salads**2 + hot_dogs**2 >= 6)
m.addConstr(fruit_salads + bowls_of_instant_ramen + hot_dogs >= 6)
m.addConstr(knishes + fruit_salads + hot_dogs >= 6)
m.addConstr(chicken_thighs + bowls_of_instant_ramen + hot_dogs >= 6)
m.addConstr(-9 * cherry_pies + 7 * hot_dogs >= 0)
m.addConstr(-7 * chicken_thighs + 10 * bowls_of_instant_ramen >= 0)
m.addConstr(8 * cherry_pies - 9 * bowls_of_instant_ramen >= 0)
m.addConstr(-7 * fruit_salads + 6 * bowls_of_instant_ramen >= 0)
m.addConstr(knishes + fruit_salads + hot_dogs <= 21)

# Solve the model
m.optimize()

# Print the results
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Cherry pies: ", cherry_pies.varValue)
    print("Knishes: ", knishes.varValue)
    print("Chicken thighs: ", chicken_thighs.varValue)
    print("Fruit salads: ", fruit_salads.varValue)
    print("Bowls of instant ramen: ", bowls_of_instant_ramen.varValue)
    print("Hot dogs: ", hot_dogs.varValue)
else:
    print("No optimal solution found")
```