Solving problems with no constraints

[andreadelprete]

Hi,

first of all, thanks for this package, it is really what I was looking for!
I'm trying to use it to optimize a simple 1d polynomial, with no constraints, but I'm getting this error:

Traceback (most recent call last):
  File "/home/student/repos/polyopt/demoPOPSolver.py", line 27, in <module>
    POP = polyopt.POPSolver(f, g, d)
  File "/home/student/repos/polyopt/polyopt/POPSolver.py", line 37, in __init__
    if max(gDegsHalf) > d:
ValueError: max() arg is an empty sequence

I guess the solver is not done for the unconstrained case. I was able to get around this by adding a dummy constraint (x+100>=0). Here is the code I'm running now:

f = {(0,): 5, (1,): -2, (2,): 1}
g = [{(0,): 1e2, (1,): 1,}] # constraint function x + 1e2 >=0
d = 2
POP = polyopt.POPSolver(f, g, d)
y0 = POP.getFeasiblePoint([array([[1]]), array([[2]]), array([[-1]]), array([[-2]])])
POP.setPrintOutput(False)
x = POP.solve(y0)

However, I still get an error:

    x = POP.solve(y0) #solve the problem
  File "/home/student/repos/polyopt/polyopt/POPSolver.py", line 80, in solve
    y = self.SDP.solve(startPoint, self.SDP.dampedNewton)
  File "/home/student/repos/polyopt/polyopt/SDPSolver.py", line 188, in solve
    x0 = method(start)
  File "/home/student/repos/polyopt/polyopt/SDPSolver.py", line 314, in dampedNewton
    FdLN = Utils.LocalNormA(Fd, Fdd)
  File "/home/student/repos/polyopt/polyopt/utils.py", line 29, in LocalNormA
    return sqrt(dot((solve(hessian, u)).T, u))[0,0]
  File "<__array_function__ internals>", line 6, in solve
  File "/home/student/.local/lib/python3.5/site-packages/numpy/linalg/linalg.py", line 399, in solve
    r = gufunc(a, b, signature=signature, extobj=extobj)
  File "/home/student/.local/lib/python3.5/site-packages/numpy/linalg/linalg.py", line 97, in _raise_linalgerror_singular
    raise LinAlgError("Singular matrix")
numpy.linalg.LinAlgError: Singular matrix

This time I'm not sure what the problem is. Any suggestion?
Moreover, would it be simple to modify the code to handle unconstrained minimization?

[andreadelprete]

Some additional info. I've tried reducing the degree of the relaxation d to 1, but nothing changed.
If I enable the output of the solver I get a lot of stuff printed, which ends with this:

k = 1035
y = [[4.27207552e+080]
 [4.51955469e+161]]
EIG[0] = [1.00000000e+000 4.51955469e+161]
EIG[1] = [4.27207552e+80]

k = 1036
y = [[4.68619274e+080]
 [5.70445470e+161]]
EIG[0] = [1.0000000e+000 5.7044547e+161]
EIG[1] = [4.68619274e+80]

k = 1037
y = [[5.60596174e+080]
 [8.12844091e+161]]

[andreadelprete]

Problem solved. Reviewing the theory I've seen that the feasible set needs to be bounded, so I added the constraint 1e4 - x^2>=0 to my problem and now the solver gives me a solution. However, very often it's not a good solution. For instance, for a polynomial with these coefficients:

a [ 0.81115 -0.45637 -0.3268   0.52828  0.24135]

I get the following solution:

Global minimum     f(x)=[42.68568] at x=[21921.42697]

While using BFGS and another scipy solver to optimize the same polynomial I get

Scalar-opt minimum f(x)=0.01851134671778265 at x=[-1.86858]
BFGS minimum        f(x)=0.564046131707772 at x=[0.62908]

I've tried playing with the degree of the relaxation, trying 1, 2, and 3, but I always get this kind of problems. Any suggestion?