How to know if the optimization problem is infeasible or not? Pyomo Warning: Problem may be infeasible - pyomo

Question

Pyomo can find a solution, but it gives this warning:
WARNING: Loading a SolverResults object with a warning status into
model=(SecondCD);
message from solver=Ipopt 3.11.1\x3a Converged to a locally infeasible point. Problem may be infeasible.
How do I know if the problem is infeasible or not?
this pyomo model optimizes a farm's decision of inputs allocation.
model.Crops = Set() # set Crops := cereal rapes maize ;
model.Inputs = Set() # set Inputs := land labor capital fertilizer;
model.b = Param(model.Inputs) # Parameters in CD production function
model.x = Var(model.Crops, model.Inputs, initialize = 100, within=NonNegativeReals)
def production_function(model, i):
return prod(model.x[i,j]**model.b[j] for j in model.Inputs)
model.Q = Expression(model.Crops, rule=production_function)
...
instance = model.create_instance(data="SecondCD.dat")
opt = SolverFactory("ipopt")
opt.options["tol"] = 1E-64
results = opt.solve(instance, tee=True) # solves and updates instance
instance.display()
if I set b >=1, (e.g.: param b := land 1 labor 1 capital 1 fertilizer 1),
pyomo can find optimal solution;
but if i set b < 1, (e.g.: param b := land 0.1 labor 0.1 capital 0.1 fertilizer 0.1), and set opt.options["tol"] = 1E-64, pyomo can find a solution, but gives that warning.
I expect an optimal solution, but the actual result gives the warning mentioned above.

Answer 1

The message you get (message from solver=Ipopt 3.11.1\x3a Converged to a locally infeasible point. Problem may be infeasible.) doesn't mean that the problem is necessarilly infeasible. A non-linear solver will typically give you a local optimum, and the path to get to the solution is a very important part of finding a "better" local optimum. When you tried with another point, you found a feasible solution, and that is the proof that your problem is feasible.
Now, in finding the global optimum instead of a local optimum, this is a little bit harder. One way to find out is to check if your problem is convex. If it is, it means that there will only be one local optimum, and that this local optimum is the global optimum. This can be done mathematically. See https://math.stackexchange.com/a/1707213/470821 and http://www.princeton.edu/~amirali/Public/Teaching/ORF523/S16/ORF523_S16_Lec7_gh.pdf from a quick Google search). If you found that your problem is not convex, then you can try to prove that there are few local optimums and that they can be found easily with good starting points. Finally, if this can't be done, you should consider more advanced techniques, all with their pros and cons. For example, you can try to generate a set of starting solutions to make sure that you cover the whole feasible domain of your problem. Another one would be to use meta-heuristics methods to help you find a better starting solution.
Also, I am sure that Ipopt have some tools to help tackling this problem of finding a good starting solution that improves the resulting local optimum.