Here's the Gurobi code to solve the optimization problem. The objective function is quadratic, and we've incorporated all the constraints provided.  Note that we've used variable names x0, x1, x2, and x3 to represent 'apple pies', 'cornichons', 'protein bars', and 'slices of pizza', respectively.

```python
from gurobipy import *

try:
    # Create a new model
    model = Model("optimization_problem")

    # Create variables
    x0 = model.addVar(lb=0, vtype=GRB.CONTINUOUS, name="apple_pies")
    x1 = model.addVar(lb=0, vtype=GRB.CONTINUOUS, name="cornichons")
    x2 = model.addVar(lb=0, vtype=GRB.INTEGER, name="protein_bars")
    x3 = model.addVar(lb=0, vtype=GRB.CONTINUOUS, name="slices_of_pizza")

    # Set objective function
    model.setObjective(9*x0*x0 + 3*x0*x1 + x0*x3 + 8*x1*x2 + x2*x2 + 9*x3*x3 + 7*x1 + 3*x2 + 2*x3, GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(10*x0 + 10*x1 + 9*x2 + 9*x3 <= 115, "protein_constraint") # r0
    model.addConstr(8*x0 + 11*x1 + 7*x2 + 6*x3 <= 96, "cost_constraint") # r1
    model.addConstr(5*x0 + 4*x1 + 4*x2 + x3 <= 102, "healthiness_constraint") # r2
    model.addConstr(x2*x2 + x3*x3 >= 14, "protein_constraint_1")
    model.addConstr(10*x1 + 9*x2 + 9*x3 >= 14, "protein_constraint_2")
    model.addConstr(10*x0 + 9*x2 + 9*x3 >= 14, "protein_constraint_3")
    model.addConstr(10*x0 + 10*x1 + 9*x3 >= 14, "protein_constraint_4")
    model.addConstr(10*x1 + 9*x2 + 9*x3 >= 21, "protein_constraint_5")
    model.addConstr(10*x0 + 9*x2 + 9*x3 >= 21, "protein_constraint_6")
    model.addConstr(10*x0 + 10*x1 + 9*x3 >= 21, "protein_constraint_7")
    model.addConstr(10*x1 + 9*x2 + 9*x3 >= 21, "protein_constraint_8")
    model.addConstr(10*x0 + 9*x2 + 9*x3 >= 21, "protein_constraint_9")
    model.addConstr(10*x0 + 10*x1 + 9*x3 >= 21, "protein_constraint_10")
    model.addConstr(7*x2*x2 + 6*x3*x3 >= 19, "cost_constraint_1")
    model.addConstr(8*x0 + 7*x2 >= 14, "cost_constraint_2")
    model.addConstr(11*x1 + 7*x2 + 6*x3 >= 21, "cost_constraint_3")
    model.addConstr(4*x2 + x3 >= 13, "healthiness_constraint_1")
    model.addConstr(100*x1*x1 + 81*x3*x3 <= 56, "protein_constraint_11") # squared protein values
    model.addConstr(10*x0 + 10*x1 <= 63, "protein_constraint_12")
    model.addConstr(100*x1*x1 + 81*x2*x2 <= 53, "protein_constraint_13") # squared protein values
    model.addConstr(10*x0 + 10*x1 + 9*x2 + 9*x3 <= 53, "protein_constraint_14")
    model.addConstr(8*x0 + 7*x2 <= 77, "cost_constraint_4")
    model.addConstr(11*x1 + 6*x3 <= 28, "cost_constraint_5")
    model.addConstr(64*x0*x0 + 121*x1*x1 <= 58, "cost_constraint_6") # squared cost values
    model.addConstr(49*x2*x2 + 36*x3*x3 <= 81, "cost_constraint_7") # squared cost values
    model.addConstr(8*x0 + 11*x1 + 7*x2 + 6*x3 <= 81, "cost_constraint_8")
    model.addConstr(4*x1 + 4*x2 <= 78, "healthiness_constraint_2")
    model.addConstr(4*x2 + x3 <= 43, "healthiness_constraint_3")
    model.addConstr(5*x0 + x3 <= 63, "healthiness_constraint_4")
    model.addConstr(4*x1 + x3 <= 25, "healthiness_constraint_5")
    model.addConstr(25*x0*x0 + 16*x2*x2 <= 85, "healthiness_constraint_6") # squared healthiness values
    model.addConstr(25*x0*x0 + 16*x2*x2 + x3*x3 <= 68, "healthiness_constraint_7") # squared healthiness values
    model.addConstr(4*x1 + 4*x2 + x3 <= 65, "healthiness_constraint_8")
    model.addConstr(25*x0*x0 + 16*x1*x1 + x3*x3 <= 71, "healthiness_constraint_9") # squared healthiness values
    model.addConstr(5*x0 + 4*x1 + 4*x2 + x3 <= 71, "healthiness_constraint_10")


    # Optimize model
    model.optimize()

    if model.status == GRB.OPTIMAL:
        print('Obj: %g' % model.objVal)
        print('Vars:')
        for v in model.getVars():
            print('%s %g' % (v.varName, v.x))
    elif model.status == GRB.INFEASIBLE:
        print("The model is infeasible.")

except GurobiError as e:
    print('Error code ' + str(e.errno) + ": " + str(e))

except AttributeError:
    print('Encountered an attribute error')
```